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Transcript
Applying Triangle Sum Properties Section 4.1 Triangles Triangles are polygons with three sides. There are several types of triangle: Scalene Isosceles Equilateral Equiangular Obtuse Acute Right Scalene Triangles Scalene triangles do not have any congruent sides. In other words, no side has the same length. 6cm 3cm 8cm Isosceles Triangle A triangle with 2 congruent sides. 2 sides of the triangle will have the same length. 2 of the angles will also have the same angle measure. Equilateral Triangles All sides have the same length Equiangular Triangles All angles have the same angle measure. Acute Triangle All angles are acute angles. Right Triangle Will have one right angle. Obtuse Angle Will have one obtuse angle. Exterior Angles vs. Interior Angles Exterior Angles are angles that are on the outside of a figure. Interior Angles are angles on the inside of a figure. Interior or Exterior? Interior or Exterior? Interior or Exterior? Triangle Sum Theorem (Postulate Sheet) States that the sum of the interior angles is 180. We will do algebraic problems using this theorem. The sum of the angles is 180, so x + 3x + 56= 180 4x + 56= 180 4x = 124 x = 31 Find the Value for X 2x + 15 2x + 15 + 3x + 90 = 180 5x + 105 = 180 3x 5x = 75 x = 15 Corollary to the Triangle Sum Theorem (Postulate Sheet) Acute angles of a right triangle are complementary. 3x + 10 5x +16 Exterior Angle Sum Theorem The measure of the exterior angle of a triangle is equal to the sum of the non-adjacent interior angles of the triangle 88 + 70 = y 158 = y 2x + 40 = x + 72 2x = x + 32 x = 32 Find x and y 46 o 8x - 1 3x + 13 2y o 4.1 Apply Congruence and Triangles 4.2 Prove Triangles Congruent by SSS, SAS 1. 2. 3. Objectives: To define congruent triangles To write a congruent statement To prove triangles congruent by SSS, SAS Congruent Polygons Congruent Triangles (CPCTC) Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent. Congruence Statement When naming two congruent triangles, order is very important. Example Which polygon is congruent to ABCDE? ABCDE -?- Properties of Congruent Triangles Example What is the relationship between C and F? D 75 F C 30 E A 75 30 B Third Angle Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. Congruent Triangles Checking to see if 3 pairs of corresponding sides are congruent and then to see if 3 pairs of corresponding angles are congruent makes a total of SIX pairs of things, which is a lot! Surely there’s a shorter way! Congruence Shortcuts? Will one pair of congruent sides be sufficient? One pair of angles? Congruence Shortcuts? Will two congruent parts be sufficient? Congruent Shortcuts? Will three congruent parts be sufficient? And if so….what three parts? Section 4.3 Proving Triangles are Congruents by SSS Draw any triangle using any 3 size lines For me I use lines of 5, 4, and 3 cm’s. Now use the same lengths and see if you can make a different triangle. 53 3cm 5cm 3cm 90 53 4cm 5cm Now measure both triangles angles and see what you get. 90 37 4cm 37 Are the following triangles congruent? Why? 10 6 6 6 6 a. 10 YES, all sides are equal so SSS 9 10 8 10 b. 9 6 No, all sides are not equal 8 ≠ 6, so fails SSS Use the SSS Congruence Postulate Decide whether the congruence statement is true. Explain your reasoning. KLM NLM SOLUTION KL NL Given KM NM Given LM LM Reflexive Property So, by the SSS Congruence Postulate, KLM NLM 4.4:Prove Triangles Congruent by SAS and HL Goal:Use sides and angles to prove congruence. Vocabulary Leg of a right triangle: In a right triangle, a side adjacent to the right angle is called a leg. Hypotenuse:In a right triangle, the side opposite the right angle is called the hypotenuse. Hypotenuse Leg Before we start…let’s get a few things straight C A Y B X INCLUDED SIDE Z Angle-Side-Angle (ASA) Congruence Postulate Two angles and the INCLUDED side Angle-Angle-Side (AAS) Congruence Postulate Two Angles and One Side that is NOT included } NO BAD WORDS Your Only Ways To Prove Triangles Are Congruent Things you can mark on a triangle when they aren’t marked. Overlapping sides are congruent in each triangle by the REFLEXIVE property Vertical Angles are congruent Alt Int Angles are congruent given parallel lines Ex 1 In ΔDEF and ΔLMN , D N , DE NL and E L. Write a congruence statement. DEF NLM Ex 2 What other pair of angles needs to be marked so that the two triangles are congruent by AAS? D E N L M F E N Ex 3 What other pair of angles needs to be marked so that the two triangles are congruent by ASA? D D L L M F E N Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 4 G K I H J ΔGIH ΔJIK by AAS Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 5 B A C D E ΔABC ΔEDC by ASA Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 6 E A C B D ΔACB ΔECD by SAS Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 7 J M K L ΔJMK ΔLKM by SAS or ASA Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 8 J T K L V Not possible U Postulate 20:Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If then Side RS UV , Angle R U , and Side RT UW , RST UVW . Example 1:Use the SAS Congruence Postulate Write a proof. Given Prove JN LN , KN MN JKN LMN J L N 1 Statements 1. JN LN , Reasons 1. Given K KN MN 2. 1 2 2. Vertical Angles Theorem 3. JKN LMN 3. SAS Congruence Postulate 2 M Example 2:Use SAS and properties of shapes In the diagram, ABCD is a rectangle. What can you conclude about By ABC the Right and Angles CDA ? Congruence Theorem, B D. Opposite sides of a rectangle are congruent, so AB CD and BC DA. ABC and CDA are congruent by the SAS Congruence Postulate. Checkpoint In the diagram, AB, CD, and EF pass through the center M of the circle. Prove that DMY BMY . Also, 1 2 3 4. Statements Reasons 1. 3 4 2. DM BM 1. Given 2. Definition of a 3. MY MY 4. DMY BMY 3. Reflexive Property of Congruence 4. SAS Congruence Postulate Checkpoint In the diagram, AB, CD, and EF pass through the center M of the circle. What can you conclude about AC and Also, 1 2 3 4. BD ? Because they are vertical angles, AMC BMD. All points on a circle are the same distance from the center, so AM BM CM DM . By the SAS Congruence Postulate, AMC BMD. Corresponding parts of congruent triangles are congruent, so you know AC BD. Theorem 4.5:Hypotenuse-Leg Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second triangle, then the two triangles are congruent. Example 3:Use the Hypotenuse-Leg Theorem Write a proof. Given AC EC , AB BD, ED BD, AC is a bisector of BD. Prove ABC EDC Example 3:Use the Hypotenuse-Leg Theorem Statements H 1. AC EC 2. AB BD, Reasons 1. Given 2. Given ED BD 3. B and D are 3. Definition of lines right angles. 4. ABC and EDC 4. Definition of a are right triangles. right triangle 5. AC is a bisector 5. Given of BD. Example 3:Use the Hypotenuse-Leg Theorem Statements L 6. BC DC Reasons 6. Definition of segment bisector 7. ABC EDC 7. HL Congruence Theorem Example 4:Choose a postulate or theorem Gate The entrance to a ranch has a rectangular gate as shown in the diagram. You know that AFC EFC. What postulate or theorem can you use to conclude that ABC EDC ? Example 4:Choose a postulate or theorem You are given that ABDE is a rectangle, so B and D are right angles. Because opposite sides of a rectangle are congruent, AB DE. You are also given that AFC EFC , so AC EC. The hypotenuse and a leg of each triangle is congruent. You can use the HL Congruence Theorem to conclude that ABC EDC. Using Congruent Triangles: CPCTC Academic Geometry Proving Parts of Triangles Congruent You know how to use SSS, SAS, ASA, and AAS to show that the triangles are congruent. Once you have triangles congruent, you can make conclusions about their other parts because, by definition, corresponding parts of congruent triangles are congruent. Abbreviated CPCTC Proving Parts of Triangles Congruent In an umbrella frame, the stretchers are congruent and they open to angles of equal measure. Given SL congruent to SR <1 congruent <2 Prove that the angles formed by the shaft and the ribs are congruent c l rib 3 4 r 1 2 stretcher s shaft Proving Parts of Triangles Congruent Prove <3 congruent <4 Statement c Reason l rib 3 4 r 1 2 stretcher s shaft Proving Parts of Triangles Congruent Given <Q congruent <R <QPS congruent <RSP p Prove SQ congruent PR Statements q Reasons r s Proving Parts of Triangles Congruent Given <DEG and < DEF are right angles. <EDG congruent <EDF Prove EF congruent EG d f e Statements Reasons g 4.7 Isosceles and Equilateral Triangles Chapter 4 Congruent Triangles 4.5 Isosceles and Equilateral Triangles Isosceles Triangle: Vertex Angle Leg Leg Base Angles Base *The Base Angles are Congruent* Isosceles Triangles Theorem 4-3 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent B <A = <C A C Isosceles Triangles Theorem 4-4 Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent B Given: <A = <C Conclude: AB = CB A C Isosceles Triangles Theorem 4-5 The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base B Given: <ABD = <CBD Conclude: AD = DC and BD is ┴ to AC A D C Equilateral Triangles Corollary: Statement that immediately follows a theorem Corollary to Theorem 4-3: If a triangle is equilateral, then the triangle is equiangular Corollary to Theorem 4-4: If a triangle is equiangular, then the triangle is equilateral Using Isosceles Triangle Theorems Explain why ΔRST is isosceles. T U Given: <R = <WVS, VW = SW Prove: ΔRST is isosceles Statement R Reason 1. VW = SW 1. Given 2. m<WVS = m<S 2. Isosceles Triangle Thm. 3. m<R = m<WVS 3. Given 4. Transitive Property 4. m<S = m<R 5. ΔRST is isosceles 5. Def Isosceles Triangle W V S Using Algebra Find the values of x and y: M ΔLMN is isosceles m<L = m< N = 63 m<LM0 = y = m<NMO 63 + 63 + y + y = 180 126 + 2y = 180 - 126 -126 2y = 54 2 2 y = 27 y° y° 27° 63° x° 63° L 27 + 63 + x = 180 90 + x = 180 -90 -90 x = 90 O N Landscaping A landscaper uses rectangles and equilateral triangles for the path around the hexagonal garden. Find the value of x. x°