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CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Unit 5: Basics of Geometry, Proofs, and Congruence After completion of this unit, you will be able to… Prove theorems about lines and angles and use them to find missing measures Prove theorems about parallel lines and use them to find missing measures Prove theorems about triangles and use them to finding missing measures Use rigid motions to determine if two figure are congruent Prove if two triangles are congruent using SAS, SSS, ASA, AAS, & HL Prove theorems about parallelograms and use them to find missing measures Timeline for Unit 2 Monday Tuesday Wednesday Thursday Friday 13 16 Day 1 – Basics of Geometry Review (cont’d) 23 Day 6 – Proving Triangle Congruency and CPCTC 17 18 Day 2 – Parallel Lines and Transversals 24 19 Day 3 – Geometric Proofs 25 Day 7 – Proving Quadrilaterals Day 4 – Triangle Angle Relationships Quiz Days 1-3 26 Day 8 – Segment Relationships in Triangles Day 1 – Basics of Geometry Review 20 Day 5 – Rigid Motions and Congruent Triangles 27 Day 9 – Review Day Day 10 – Unit 5 Test 1 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Day 1 – Basics of Geometry Naming Angles and Lines Line Segment Point Points are named with capital letters. Two points are connected with a straight line. This line segment can be named AB or BA . Line Ray A line does not have a beginning or end point. Lines are named using two points on the line. Rays start with a point but continue to infinity in one direction. Rays are named using its starting point and one other point on the ray. The ray This line can be named VW or WV . Angle can be named AB but NOT BA . Angles are made up of two rays that have the same beginning point. The point is called the vertex and the two rays are called the side of the angle. Angles can be name in ways: One Letter (if the vertex is not shared): A Number (if given): 1 Three Letters (vertex is middle letter): MAS or SAM Practice: Name the following angles: a. Name the angle in four ways: b. Name angle 1 as many ways as possible: 2 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes TYPES OF ANGLES Acute Angles Acute angles have measures between ____ & ____ Obtuse Angles Obtuse Angles have measures between ____ & ____ Right Angles Right Angles measure exactly ____ Straight Angles Straight Angles measure exactly ____ Important Geometry Symbols Angle Triangle Congruent (same shape & size) Degrees Perpendicular (90 degrees) m Measure of Parallel Congruent Angles Similar ` Congruent Segments Using Geometry Terminology A conditional statement (if-then) is a statement that contains a hypothesis (if) and conclusion (then). Ex. If a student plays basketball, then they are an athlete. A converse is a statement that has the hypothesis and conclusion switched around. Ex. If a student is an athlete, then they play basketball. (Is this true?) A postulate is a statement that is accepted as true without proof. A theorem is a statement that must be proven before it can be accepted as true. We are going to prove many theorems throughout this unit. We will prove a few of the following relationships on Day 3. Practice: Take the following statement: I do my homework; I get my allowance, and write it in if-then form and then write the converse of it. 3 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Supplementary and Complementary Angles Complementary Angles: Two or more angles whose sum of measures equals 90°. 40° and 50° angles are complementary angles because 40° + 50° = 90°. Complementary Angles Example: A 30° angle is called the complement of the 60° angle. Similarly, the 60° angle is the complement of the 30° angle. Practice: Find the complement of each angle. a. 35° b) Two angles, 2x° and 3x° are complementary. Find the value of x and each angle. Supplementary Angles: Two or more angles whose sum of measures equals 180°. 60° and 120° angles are supplementary angles because 60° + 120° = 180°. Supplementary Angles Example: A 70° angle is called the supplement of the 110° angle. Similarly, the 110° angle is the supplement of the 70° angle. Practice: Find the supplement of each angle. a.) 126° b) Two angles, 4x° and 6x° are supplementary. Find the value of x and each angle. 4 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Special Pairs of Angles Linear Pair: Two adjacent (next to) angles whose noncommon sides are opposite rays. A linear pair also forms a line (supplementary). Linear Pair a. Name all the linear pairs in the diagram below: b. Solve for x: Vertical Angles: Two nonadjacent angles that are formed by two intersecting lines. Vertical angles are congruent. Vertical Angles a. Name all the vertical angles in the diagram below: b. Find the measure of angles 1, 2, 3, and 4. c. Solve for x. Then determine the measure of angle 1. 5 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Angle Relationships Angle Addition Postulate: If point D lies in the interior of a. Find the measure of ABC, then m ABD + m DBC = m ABC. PTM: Angle Addition b. Given mÐQST = 135 , find mÐQSR . Perpendicular: Two lines, rays, or segments that intersect to form a 90° angle. a. Name all the angles you know are right angles. Perpendicular b. Solve for x. 6 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Angle Bisector: A ray that divides an angle into two congruent angles (two angles with equal measure). a. QS bisects PQR . Find mPQS . b. KM bisects JKL . Find the value of x. Angle Bisector Segment Relationships Segment Addition Postulate: If point B is on AC , and between points A and C, then AB + BC = AC . Segment Addition a. Use the diagram to find EF . b. Write an expression for AC. c. Find the value of z. 7 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Midpoint: Point that divides the segment into two congruent segments. Midpoint a. Find FM and MG . b. Find YM and YZ . c. T is the midpoint of QR . Solve for x. Segment Bisector: A line, line segment, or ray that divides the line segment into two line segments of equal length. Segment Bisector a. Find CB and AB . b. Determine if you have enough information to determine if PC is the segment bisector of AB . Explain why or why not. Perpendicular Bisector: A line, line segment, or ray that intersects at the midpoint of a line segment at a 90 degree angle. a. Determine if you have enough information to determine if WY is the perpendicular bisector of ZX . Explain why or why not. Perpendicular Bisector 8 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Day 2 – Parallel Lines Alternate Exterior Angles: Alternate Interior Angles: Same Side (Consecutive) Exterior Angles: Same Side (Consecutive) Interior Angles: Corresponding Angles: Alternate Exterior Angles: Two angles in the exterior of the parallel lines and on opposite sides. Alternate Interior Angles: Two angles in the interior of the parallel lines and on opposite sides. Same Side Exterior Angles: Two angles in the exterior of the parallel lines and on same sides. Same Side Interior Angles: Two angles in the interior of the parallel lines and on same sides. Corresponding Angles: Two angles that lie in the same relative location. A transversal is a line that intersects two or more coplanar lines at different points. Postulates and Theorems Regarding Parallel Lines Cut by a Transversal: Corresponding Angles Postulate If 2 PARALLEL LINES are cut by a transversal, then the pairs of corresponding angles are congruent. Alternate Interior Angles Theorem If 2 PARALLEL LINES are cut by a transversal, then the pairs of alternate interior angles are congruent. 9 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Alternate Exterior Angles Theorem Consecutive Interior Angles Theorem If 2 PARALLEL LINES are cut by a transversal, then the pairs of alternate exterior angles are congruent. If 2 PARALLEL LINES are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Summary of Parallel Line Relationships Relationships with Parallel & Non Parallel Lines Angle Type Alternate Exterior Angles Parallel Lines Non Parallel Lines Alternate Interior Angles Same Side Exterior Angles Same Side Interior Angles Corresponding Angles Vertical Angles Practice: 1. If the measure of angle 1 = 67 and a d, find the measure of all other angles. 10 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes 2. Find the measure of the following: a. Solve for x: b. mECF c. mDCE 3. Find the measure of the following: a. Solve for x: b. mEDG c. mBDG 11 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Day 3 – Intro to Proofs (Algebraic and Geometric) 1. Solve the following equation. Justify each step as you solve it. 2(4x - 3) – 8 = 4 + 2x 2. Rewrite your proof so it is “formal” proof. 2(4x - 3) – 8 = 4 + 2x When writing a geometric proof, you create a chain of logical steps that move from the hypothesis to the conclusion of the conjecture you are proving. By proving the conclusion is true, you have proven the original conjecture is true. When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them. Two Column Proofs ______________________________________________ ______________________________________________ ______________________________________________ 12 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Properties of Equality and Other Definitions Property General Example Example 1 Addition Property If a = b, then a + c = b + c If x – 4 = 8, then x = 12 Subtraction Property If a = b, then a – c = b - c If x + 5 = 7, then x = 2 Multiplication Property If a = b, then ac = bc If = 9, then x = 18 Division Property If a = b, 𝑎 𝑏 then = If 2x = 10, then x = 5 Distributive Property If a(b + c), then ab + ac If 2 (x + 5), then 2x + 10 Reflexive Property a=a 5=5 ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐴𝐵 Symmetric Property If a = b, then b = a If 2 = x, then x = 2 If ∠𝐴 ≅ ∠𝐵, then ∠𝐵 ≅ ∠𝐴 Transitive Property If a = b, and b = c, then a = c ̅̅̅̅ , If ̅̅̅̅̅ 𝑊𝑋 ≅ ̅̅̅̅ 𝑋𝑌 and ̅̅̅̅ 𝑋𝑌 ≅𝑌𝑍 ̅̅̅̅̅ ̅̅̅̅ then 𝑊𝑋 ≅𝑌𝑍 Substitution Property If x = y, then y can be substituted for x in any expression If ̅̅̅̅ 𝐴𝐵 + ̅̅̅̅ 𝐵𝐶 = ̅̅̅̅ 𝑋𝑍 and ̅̅̅̅ 𝐴𝐵 = ̅̅̅̅ ̅̅̅̅ 𝐶𝐷 , then 𝐶𝐷 + ̅̅̅̅ 𝐵𝐶 = ̅̅̅̅ 𝑋𝑍 𝑐 𝑐 Example 2 𝑥 2 Definition of Congruent Segments If mA mB , then A B . Definition of Congruent Angles If mAB mCD , then AB CD . If ∠A ≅ ∠B and ∠B ≅∠C, then ∠A ≅ ∠C If ∠A is supplementary to ∠B and ∠B ≅ ∠C, then ∠A is supplementary to ∠C Algebraic Proofs Practice #1: STATEMENTS REASONS PROVE: AB = 13 13 CCGPS Geometry Practice #2: Unit 5: Intro to Geometry and Congruence STATEMENTS Notes REASONS PROVE: x = Practice: #3 STATEMENTS REASONS Prove: x = 14 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Geometric Proofs Prove the Linear Pair Theorem by filling in the blanks of a two column proof. Given: Angle 1 and 2 form a linear pair. Prove: Angle 1 and 2 are supplementary. Statements 1. 1 and 2 form a linear pair. Reasons 1. Given 2. BA and BC form a line. 2. __________________________________________ 3. m ABC = 180 3. __________________________________________ 4. ______________________________________ 4. Angle Addition Postulate 5. ______________________________________ 5. Substitution 6. 1 and 2 are supplementary 6. _________________________________________ Prove the Vertical Angles Theorem using a two column proof. Given: Angle 4 and 1 are a linear pair. Angle 1 and 2 are a linear pair. Prove: 2 4 1. Statements Reasons 1. Given 2. 2. Given 3. 3. Linear Pair Theorem 4. 4. Linear Pair Theorem 5. 5. Definition of Supplementary Angles 6. 6. Definition of Supplementary Angles 7. 7. Substitution 8. 8. Subtraction Property 9. 9. Definition of Congruent Angles 15 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Prove the Alternate Interior Angles are Congruent Theorem using a two column proof: Given : l m Pr ove : 2 3 Statements 1. ______________________________ Reasons 1. Given 2. ______________________________ 2. Vertical Angles are Congruent 3. ______________________________ 3. Corresponding Angles Postulate 4. ______________________________ 4. Transitive Property Prove the Right Angle Congruence Theorem using a two column proof. Given: ACD and BCD are right angles Prove: ACD BCD Statements 1. ___________________________ Reasons 1. Given 2. ___________________________ 2. ___________________________ 3. mACD 90 3. ___________________________ 4. ___________________________ 4. ___________________________ 5. ___________________________ 5. Transitive Property 6. ACD BCD 6. ___________________________ Prove the Congruent Supplement Theorem using a two column proof: Given: Angle 1 is supplementary to angle 2 Angle 3 is supplementary to angle 4 2 4 Prove: 1 3 Statements 1. ______________________________ Reasons 1. Given 2. ______________________________ 2. Given 3. 2 4 3. ____________________________________ 4. ______________________________ 4. Definition of congruent angles 5. ______________________________ 5. Defintion of supplementary angles 6. m3 m4 180 6. ____________________________________ 7. ______________________________ 7. Substitution Property 8. m1 m2 m3 m2 8. ____________________________________ 9. ______________________________ 9. Subtraction Property of Equality 10. ______________________________ 10. ____________________________________ 16 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Day 4 – Angle Relationships in Triangles A triangle is a figure formed when three noncollinear (not on the same line) points are connected by segments. The sides are: Opposite Side of F: The vertices are: Opposite Side of E: The angles are: Opposite Side of D: Triangles can be classified by two categories: by Angles and by Sides. Acute All Acute Angles Scalene No Sides Congruent ANGLES Obtuse One Obtuse Angle Right One Right Angle SIDES Isosceles Equilateral At Least 2 Sides Congruent All Sides Congruent Practice: Classify the triangles by sides and angles. Think About It: Check which triangles are possible. Acute Obtuse Right Scalene Isosceles Equilateral 17 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Triangle Sum Theorem Triangle Sum Theorem: The measures of the three interior angles in a triangle add up to be 180 This means: Corollary to Triangle Sum Theorem: The acute angles of a right triangle are complementary. This means: Proof of the Triangle Sum Theorem: Statements 1. _____________________________ 1. Given 2. _____________________________ 2. Given 3. 4 1 3. ____________________________ 4. _____________________________ 4. Alt. Interior Angles Theorem 5. m4 m1 5. ____________________________ 6. _____________________________ 6. Def. of Congruent Angles 7. _____________________________ 7. Linear Pair Postulate 8. mACD m5 m3 8. ____________________________ 9. _____________________________ 9. Substitution Property 10. _____________________________ Examples: Find m U. Reasons Find m UPM 10. ____________________________ Find the measure of x. 18 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Side Inequality Theorem Side Inequality Theorem: If one side of a triangle is longer than the other side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. This means: The largest angle of a triangle lies opposite the longest side. The smallest angle lies opposite the shortest side. If two angles are equal, their side lengths will be equal. Example: List the sides from shortest to longest for each diagram. Exterior Angle Theorem Exterior Angle Theorem: The measure of the exterior angle is equal to the sum of two remote interior angles. Interpret: What does exterior mean? __________________________________ What does interior mean? ___________________________________ What does remote mean? ___________________________________ Proof: Prove the exterior angle theorem: Given: 1, 2, and 3 are interior angles. Prove: 4 = 1 + 2 19 CCGPS Geometry Examples: A. Unit 5: Intro to Geometry and Congruence B. Notes C. Solve for x using the Exterior Angle Theorem: a. b. c. Solve for x using the Triangle Sum Theorem: a. b. c. 20 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Isosceles Base Angle Theorem and Its Converse Examples: A. Find the value of x D. Find the measure of <P. R = 30 Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent. Converse of Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite of them are congruent. B. Find the m T C. Find the value of x. E. Find the measure of Q F. Find the value of x & y. 21 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Day 5 – Rigid Motions and Congruent Triangles Translations: Type of transformation in which a figure “slides” horizontally, vertically, or both. Rule: (x, y) (x + a, y + b) a left or right translations b up or down translations A. Graph triangle ABC by plotting points A(8, 10), B(1, 2), and C(8, 2). B. Translate triangle ABC 15 units to the left to form triangle A’B’C’ and write new coordinates. C. Translate triangle ABC 12 units down to form triangle A’’B’’C’’ and write new coordinates. Coordinates of Triangle ABC Coordinates of Triangle A’B’C’ Coordinates of Triangle A’’B’’C’’ A (8, 10) B (1, 2) C(8, 2)) Observation: Did the figure change size or shape after each translation? Reflections: Type of transformation in which a figure “flips” over a line of symmetry. x-axis: (x, y) (x, -y) y –axis (x, y) (-x, y) A. Graph triangle ABC by plotting points A(8, 10), B(1, 2), and C(8, 2). B. Reflect triangle ABC over the x-axis to form triangle A’B’C’ and write new coordinates. C. Reflect triangle ABC over the y-axis to form triangle A’’B’’C’’ and write new coordinates. Coordinates of Triangle ABC Coordinates of Triangle A’B’C’ Coordinates of Triangle A’’B’’C’’ A (8, 10) B (1, 2) C(8, 2)) Observation: Did the figure change size or shape after each reflection? 22 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Rotations: Type of transformation in which a figure “turns” about a fixed point at a given angle and direction. 90 clockwise or 270 counterclockwise: (x, y) (y, -x) 180 clockwise or 180 counterclockwise: (x, y) (-x, -y) 270 clockwise or 90 counterclockwise: (x, y) (-y, x) A. Graph triangle ABC by plotting points A(8, 10), B(1, 2), and C(8, 2). B. Rotate triangle ABC 90 clockwise to form triangle A’B’C’ and write new coordinates. C. Rotate triangle ABC 180 counterclockwise to form triangle A’’B’’C’’ and write new coordinates. D. Rotate triangle ABC 90 counterclockwise to form triangle A’’’B’’’C’’’ and write new coordinates. Coordinates of Triangle ABC Coordinates of Triangle A’B’C’ Coordinates of Triangle A’’B’’C’’ Coordinates of Triangle A’’’B’’’C’’’ A (8, 10) B (1, 2) C(8, 2)) Observation: Did the figure change size or shape after each rotation? What you just discovered is that certain transformations preserve the size and shape of figures. Transformations that preserve size and shape are called rigid motions or isometries. When two figures maintain their size and shape, they remain congruent to each other. Any combination of translations, rotations, and reflections of a figure will ALWAYS maintain congruency to the original figure. Two triangles are congruent to each other if and only if their corresponding angles and sides are congruent. Corresponding parts of triangles are the parts of the congruent triangles that “match.” Example: If RST XYZ , identify all pairs of congruent corresponding parts. Draw a picture and label the congruent angles and sides. Third Angle Theorem: If two angles on one triangle are congruent to two angles to another triangle, then the third pair of angles are congruent. Example: Find the measures of angles C and I. 23 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Triangle Congruence Theorems Side – Side – Side (SSS) Congruence Theorem three sides of one triangle are congruent to three sides of a second triangle Angle – Side – Angle (ASA) Congruence Theorem two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle Side – Angle – Side (SAS) Congruence Theorem two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle Angle – Angle – Side (AAS) Congruence Theorem two angles and a non-included side of one triangle are congruent to two angles and a non-included side of a second triangle Hypotenuse – Leg (HL) Congruence Theorem (RIGHT TRIANGLES ONLY!) In a right triangle, the hypotenuse and one leg is congruent to the hypotenuse and leg of another right triangle Included Side The side between two angles Included Angle The angle between two sides Practice: Mark the included side in each triangle Practice: Mark the included angle in each triangle. 24 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Practice: Mark the appropriate sides and angles to make each congruence statement true by the stated congruence theorem. a. ASA b. SSS c. SAS d. HL e. AAS f. Your Choice: _________ Markings You Are Allowed to Add Share a Side Reason: Reflexive Property Vertical Angles Reason: Vertical Angles are Congruent 25 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Practice: Determine whether there is enough information to conclude if the triangles are congruent. If so, state the congruence theorem. If not, write not enough information. A. B. C. D. E. F. . G. H. I. J. K. L. 26 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Day 6 – Proving Triangles Congruent (including CPCTC) From yesterday, you learned that you only need 3 pieces of information (combination of angles and sides) to determine if two triangles are congruent. Today, we are going to prove two triangles are congruent using two column proofs. A. Given: AB CD, BC AD Prove: ABC CDA ? B. C. Given: AB CD and AE CE Pr ove : ABE CDE? Given : RF BP and BF RP Pr ove : RFP BPF? 27 CCGPS Geometry D. Given : WN HK Pr ove : WNZ HKZ? Unit 5: Intro to Geometry and Congruence E. Given: JA MY and YM bi sects JYA Pr ove :JYM AYM? F. Given: JA MY and JY AY Pr ove :JYM AYM? Notes 28 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes CPCTC Once you conclude two triangles are congruent, then you can also conclude that corresponding parts of congruent triangles are congruent (CPCTC). CPCTC can be used as a justification after you have proved two triangles are congruent. Look at the example proof below: Statements Reasons 1. SU SK 1. Given 2. SR SH 2. Given 3. S S 3. Reflexive Property 4. SUH SKR 4. SAS 5. U K 5. CPCTC Prove the Isosceles Base Angles Theorem Given: GB GD , GU bis ec ts DGB Prove: B D Statements Reasons 2. _________________________ 1. Given 2. _________________________ 3. _________________________ 3. Definition of bisects 4. GU GU 4. _________________________ 5. _________________________ 5. _________________________ 6. B D 6. _________________________ 1. GB GD Prove the Converse of the Isosceles Base Angles Theorem: Given: B D , GU bis ec ts DGB Prove: GB GD Statements 1. B D Reasons 2. _________________________ 1. Given 2. _________________________ 3. _________________________ 3. _________________________ 4. GU GU 4. _________________________ 5. _________________________ 5. _________________________ 6. GB GD 6. _________________________ 29 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Proof: Given: JHK LHK, JKH LKH Prove: JK LK Proof: Given: CW and SD bi sect each other Prove: CS WD Proof: Given: AB CB, D is the midpoint of AC Prove: A C 30 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Day 7 – Segment Relationships in Triangles A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which forms the midsegment triangle. Triangle Midsegment Theorem: A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. The Midsegment is: Parallel to one side of the triangle Is half the length of the parallel side Connects to the midpoints Practice: A. Solve for x: D. Given CD = 14, GF = 8, and GC = 5, Find the perimeter of BCD . Midsegments: Midsegment Triangle: B. Solve for x: C. Solve for x, y, and z: E. Find the measure of the following: JL PM MLK 31 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Perpendicular Bisectors of Triangles If you remember from Day 1, perpendicular bisectors are lines, line segments, or rays that intersect at the midpoint of a line segment at a 90 degree angle. Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If: Then: Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment. Practice: A. Find BC. B. Find AD if AC is the perpendicular bisector to BD. C. Find TU 32 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Points of Concurrency When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect. You will examine several different points of concurrency today. Medians The median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Centroid The centroid is where the three medians of a triangle intersect. The centroid is always located on the inside of the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. Angle Bisectors A triangle has three angle bisectors. Angle bisectors divide an angle into two congruent angles. Incenter The incenter is where the three angle bisectors of a triangle intersect. The incenter is always located inside the triangle. The incenter is located equidistant from the sides of the triangle. PX = PY = PZ 33 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Perpendicular Bisectors A triangle has three perpendicular bisectors. They pass through the midpoint on each side of a triangle at a 90 angle. Notes Circumcenter The circumcenter is where the three perpendicular bisectors of a triangle intersect. The circumcenter is located equidistant from the vertices of the triangle. PA = PB = PC The circumcenter can be located inside (acute), outside (obtuse) or on (right) a triangle. Altitudes Orthocenter A triangle has three altitudes. An altitude is a perpendicular segment from a vertex to the line containing the opposite side. The othrocenter is where the three altitudes of a triangle intersect. The orthocenter is located inside for an acute triangle, outside for an obtuse triangle, and on the vertex for a right triangle. A triangle’s altitude can be located in three locations (inside, on, or outside) the triangle. 34 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes The Incenter and Circumcenter also represent the center of a circle inscribed or circumscribed with a triangle. Incenter with an Inscribed Circle (Inscribed – circle is inside the triangle)) Circumcenter with a Circumscribed Circle (Circumscribed – circle is outside the triangle) How Can I Remember How the Points of Concurrency Are Created? Aunt Betty Insists Peanut Butter Cookies Are Only Made Centrally Angle Bisector – Incenter Perpendicular Bisector – Circumcenter Altitudes – Orthocenter Medians – Centroid Practice: Name the point of concurrency: a. b. e. Use the diagram as shown; G is the _____________________ of triangle ABC. If GC = 2x – 7 and GB = -3x + 13, what is the value of GA? c. d. f. Use the diagram as shown; G is the _____________________ of triangle ABC. If GD = 4x + 8 and GE = 7x - 10, what is the value of x? 35 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Day 8 – Parallelograms A parallelogram is a type of quadrilateral that has two pairs of opposite sides that are parallel. Parallelograms are denoted by the symbol . If a quadrilateral has two pairs of opposite sides, then it can be classified as a parallelogram. There are 5 theorems associated with parallelograms: Opposite sides are congruent Opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other Diagonals form two congruent triangles Theorem 1: Opposite sides are congruent Theorem 2: Opposite angles are congruent. Theorem 3: Consecutive angles are supplementary. Find the value of x. Then find the length of BC. Find the value of x. Then find Angle Q. Find the value of y. Then find the measure of angle A and B. 36 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Theorem 4: Diagonals bisect each other. Notes EFGH is a parallelogram. Find w and z. Given: JKLM is a parallelogram Prove: JKL LMJ Theorem 5: Diagonals form two congruent triangles. Practice: Apply the properties of parallelograms to solve for the specified variable. a. Solve for x, y, and z. b. Solve for x, y, and z. c. Solve for x and z. 37 CCGPS Geometry Unit 5: Intro to Geometry and Congruence Notes Proofs with Parallelograms Determine if each quadrilateral must be a parallelogram. Explain why or why not. a. b. c. Given: ABCD is a parallelogram Statements Prove: BAD DCB 1. ABCD is a parallelogram Reasons 1. ___________________________ 2. AB CD 2. ___________________________ 3. DA BC 3. ___________________________ 4. ___________________________ 4. ___________________________ 5. ___________________________ 5. SSS 6. ___________________________ 6. ___________________________ d. Given: ABCD and AFGH are parallelograms Prove: C G Statements 1. ABCD is a parallelogram Reasons 1. ___________________________ 2. ___________________________ 2. Given 3. C A 3. ___________________________ 4. A G 4. ___________________________ 5. ___________________________ 5. ____________________________ 38