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Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 Lesson 4-1 Classifying Triangles Lesson 4-2 Angles of Triangles Lesson 4-3 Congruent Triangles Lesson 4-4 Proving Congruence–SSS, SAS Lesson 4-5 Proving Congruence–ASA, AAS Lesson 4-6 Isosceles Triangles Lesson 4-7 Triangles and Coordinate Proof Example 1 Classify Triangles by Angles Example 2 Classify Triangles by Sides Example 3 Find Missing Values Example 4 Use the Distance Formula ARCHITECTURE The triangular truss below is modeled for steel construction. Classify JMN, JKO, and OLN as acute, equiangular, obtuse, or right. Answer: JMN has one angle with measure greater than 90, so it is an obtuse triangle. JKO has one angle with measure equal to 90, so it is a right triangle. OLN is an acute triangle with all angles congruent, so it is an equiangular triangle. ARCHITECTURE The frame of this window design is made up of many triangles. Classify ABC, ACD, and ADE as acute, equiangular, obtuse, or right. Answer: ABC is acute. ACD is obtuse. ADE is right. Identify the isosceles triangles in the figure if Isosceles triangles have at least two sides congruent. Answer: UTX and UVX are isosceles. Identify the scalene triangles in the figure if Scalene triangles have no congruent sides. Answer: VYX, ZTX, VZU, YTU, VWX, ZUX, and YXU are scalene. Identify the indicated triangles in the figure. a. isosceles triangles Answer: ADE, ABE b. scalene triangles Answer: ABC, EBC, DEB, DCE, ADC, ABD ALGEBRA Find d and the measure of each side of equilateral triangle KLM if and Since KLM is equilateral, each side has the same length. So Substitution Subtract d from each side. Add 13 to each side. Divide each side by 3. Next, substitute to find the length of each side. Answer: For KLM, and the measure of each side is 7. ALGEBRA Find d and the measure of each side of equilateral triangle if and Answer: COORDINATE GEOMETRY Find the measures of the sides of RST. Classify the triangle by sides. Use the distance formula to find the lengths of each side. Answer: ; since all 3 sides have different lengths, RST is scalene. Find the measures of the sides of ABC. Classify the triangle by sides. Answer: ; since all 3 sides have different lengths, ABC is scalene. Example 1 Interior Angles Example 2 Exterior Angles Example 3 Right Angles Find the missing angle measures. Find first because the measure of two angles of the triangle are known. Angle Sum Theorem Simplify. Subtract 117 from each side. Angle Sum Theorem Simplify. Subtract 142 from each side. Answer: Find the missing angle measures. Answer: Find the measure of each numbered angle in the figure. Exterior Angle Theorem Simplify. If 2 s form a linear pair, they are supplementary. Substitution Subtract 70 from each side. Exterior Angle Theorem Substitution Subtract 64 from each side. If 2 s form a linear pair, they are supplementary. Substitution Simplify. Subtract 78 from each side. Angle Sum Theorem Substitution Simplify. Subtract 143 from each side. Answer: Find the measure of each numbered angle in the figure. Answer: GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20. Corollary 4.1 Substitution Subtract 20 from each side. Answer: The piece of quilt fabric is in the shape of a right triangle. Find if is 32. Answer: Example 1 Corresponding Congruent Parts Example 2 Transformations in the Coordinate Plane ARCHITECTURE A tower roof is composed of congruent triangles all converging toward a point at the top. Name the corresponding congruent angles and sides of HIJ and LIK. Answer: Since corresponding parts of congruent triangles are congruent, ARCHITECTURE A tower roof is composed of congruent triangles all converging toward a point at the top. Name the congruent triangles. Answer: HIJ LIK The support beams on the fence form congruent triangles. a. Name the corresponding congruent angles and sides of ABC and DEF. Answer: b. Name the congruent triangles. Answer: ABC DEF COORDINATE GEOMETRY The vertices of RST are R(─3, 0), S(0, 5), and T(1, 1). The vertices of RST are R(3, 0), S(0, ─5), and T(─1, ─1). Verify that RST RST. Use the Distance Formula to find the length of each side of the triangles. Use the Distance Formula to find the length of each side of the triangles. Use the Distance Formula to find the length of each side of the triangles. Answer: The lengths of the corresponding sides of two triangles are equal. Therefore, by the definition of congruence, Use a protractor to measure the angles of the triangles. You will find that the measures are the same. In conclusion, because , COORDINATE GEOMETRY The vertices of RST are R(─3, 0), S(0, 5), and T(1, 1). The vertices of RST are R(3, 0), S(0, ─5), and T(─1, ─1). Name the congruence transformation for RST and RST. Answer: RST is a turn of RST. COORDINATE GEOMETRY The vertices of ABC are A(–5, 5), B(0, 3), and C(–4, 1). The vertices of ABC are A(5, –5), B(0, –3), and C(4, –1). a. Verify that ABC ABC. Answer: Use a protractor to verify that corresponding angles are congruent. b. Name the congruence transformation for ABC and ABC. Answer: turn Example 1 Use SSS in Proofs Example 2 SSS on the Coordinate Plane Example 3 Use SAS in Proofs Example 4 Identify Congruent Triangles ENTOMOLOGY The wings of one type of moth form two triangles. Write a two-column proof to prove that FEG HIG and G is the midpoint of both Given: G is the midpoint of both Prove: FEG HIG Proof: Statements Reasons 1. 1. Given 2. 2. Midpoint Theorem 3. FEG HIG 3. SSS Write a two-column proof to prove that ABC Proof: Statements Reasons 1. 2. 3. ABC GBC 1. Given 2. Reflexive 3. SSS GBC if COORDINATE GEOMETRY Determine whether WDV MLP for D(–5, –1), V(–1, –2), W(–7, –4), L(1, –5), P(2, –1), and M(4, –7). Explain. Use the Distance Formula to show that the corresponding sides are congruent. Answer: By definition of congruent segments, all corresponding segments are congruent. Therefore, WDV MLP by SSS. Determine whether ABC DEF for A(5, 5), B(0, 3), C(–4, 1), D(6, –3), E(1, –1), and F(5, 1). Explain. Answer: By definition of congruent segments, all corresponding segments are congruent. Therefore, ABC DEF by SSS. Write a flow proof. Given: Prove: QRT STR Answer: Write a flow proof. Given: Prove: ABC ADC . Proof: Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. Two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. The triangles are congruent by SAS. Answer: SAS Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. Each pair of corresponding sides are congruent. Two are given and the third is congruent by Reflexive Property. So the triangles are congruent by SSS. Answer: SSS Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. a. Answer: SAS b. Answer: not possible Example 1 Use ASA in Proofs Example 2 Use AAS in Proofs Example 3 Determine if Triangles Are Congruent Write a paragraph proof. Given: L is the midpoint of Prove: WRL EDL Proof: because alternate interior angles are congruent. By the Midpoint Theorem, Since vertical angles are congruent, WRL EDL by ASA. Write a paragraph proof. Given: Prove: ABD CDB Proof: because alternate interior angles are congruent. because alternate interior angles are congruent. by Reflexive Property. ABD CDB by ASA. Write a flow proof. Given: Prove: Proof: Write a flow proof. Given: Prove: Proof: STANCES When Ms. Gomez puts her hands on her hips, she forms two triangles with her upper body and arms. Suppose her arm lengths AB and DE measure 9 inches, and AC and EF measure 11 inches. Also suppose that you are given that Determine whether ABC EDF. Justify your answer. Explore We are given measurements of two sides of each triangle. We need to determine whether the two triangles are congruent. Plan Since Likewise, We are given Check each possibility using the five methods you know. Solve We are given information about three sides. Since all three pairs of corresponding sides of the triangles are congruent, ABC EDF by SSS. Examine You can measure each angle in ABC and EDF to verify that Answer: ABC EDF by SSS The curtain decorating the window forms 2 triangles at the top. B is the midpoint of AC. inches and inches. BE and BD each use the same amount of material, 17 inches. Determine whether ABE CBD Justify your answer. Answer: ABE CBD by SSS Example 1 Proof of Theorem Example 2 Find the Measure of a Missing Angle Example 3 Congruent Segments and Angles Example 4 Use Properties of Equilateral Triangles Write a two-column proof. Given: Prove: Proof: Statements Reasons 1. 1. Given 2. 2. Def. of 3. ABC and BCD are isosceles 4. 3. Def. of isosceles 5. 6. 5. Given 6. Substitution segments 4. Isosceles Theorem Write a two-column proof. Given: Prove: . Proof: Statements 1. Reasons 1. Given 2. ADB is isosceles. 2. Def. of isosceles triangles 3. 4. 3. Isosceles Theorem 4. Given 5. 6. ABC ADC 7. 5. Def. of midpoint 6. SAS 7. CPCTC Multiple-Choice Test Item If and measure of A. 45.5 B. 57.5 Read the Test Item CDE is isosceles with base isosceles with what is the C. 68.5 D. 75 Likewise, CBA is Solve the Test Item Step 1 The base angles of CDE are congruent. Let Angle Sum Theorem Substitution Add. Subtract 120 from each side. Divide each side by 2. Step 2 are vertical angles so they have equal measures. Def. of vertical angles Substitution Step 3 The base angles of CBA are congruent. Angle Sum Theorem Substitution Add. Subtract 30 from each side. Divide each side by 2. Answer: D Multiple-Choice Test Item If and measure of A. 25 Answer: A B. 35 what is the C. 50 D. 130 Name two congruent angles. Answer: Name two congruent segments. By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So, Answer: a. Name two congruent angles. Answer: b. Name two congruent segments. Answer: EFG is equilateral, and Find and bisects bisects Each angle of an equilateral triangle measures 60°. Since the angle was bisected, is an exterior angle of EGJ. Exterior Angle Theorem Substitution Add. Answer: EFG is equilateral, and Find bisects bisects Linear pairs are supplementary. Substitution Subtract 75 from each side. Answer: 105 ABC is an equilateral triangle. a. Find x. Answer: 30 b. Answer: 90 bisects Example 1 Position and Label a Triangle Example 2 Find the Missing Coordinates Example 3 Coordinate Proof Example 4 Classify Triangles Position and label right triangle XYZ with leg long on the coordinate plane. d units Use the origin as vertex X of the triangle. Place the base of the triangle along the positive x-axis. Position the triangle in the first quadrant. Since Z is on the x-axis, its y-coordinate is 0. Its X (0, 0) x-coordinate is d because the base is d units long. Z (d, 0) Since triangle XYZ is a right triangle the x-coordinate of Y is 0. We cannot determine the y-coordinate so call it b. Answer: Y (0, b) X (0, 0) Z (d, 0) Position and label equilateral triangle ABC with side w units long on the coordinate plane. Answer: Name the missing coordinates of isosceles right triangle QRS. Q is on the origin, so its coordinates are (0, 0). The x-coordinate of S is the same as the x-coordinate for R, (c, ?). The y-coordinate for S is the distance from R to S. Since QRS is an isosceles right triangle, The distance from Q to R is c units. The distance from R to S must be the same. So, the coordinates of S are (c, c). Answer: Q(0, 0); S(c, c) Name the missing coordinates of isosceles right ABC. Answer: C(0, 0); A(0, d) Write a coordinate proof to prove that the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base is perpendicular to the base. The first step is to position and label a right triangle on the coordinate plane. Place the base of the isosceles triangle along the x-axis. Draw a line segment from the vertex of the triangle to its base. Label the origin and label the coordinates, using multiples of 2 since the Midpoint Formula takes half the sum of the coordinates. Given: XYZ is isosceles. Prove: Proof: By the Midpoint Formula, the coordinates of W, the midpoint of , is The slope of or undefined. The slope of therefore, is . Write a coordinate proof to prove that the segment drawn from the right angle to the midpoint of the hypotenuse of an isosceles right triangle is perpendicular to the hypotenuse. Proof: The coordinates of the midpoint D are The slope of or 1. The slope of therefore . is or –1, DRAFTING Write a coordinate proof to prove that the outside of this drafter’s tool is shaped like a right triangle. The length of one side is 10 inches and the length of another side is 5.75 inches. Proof: The slope of or undefined. The slope of or 0, therefore DEF is a right triangle. The drafter’s tool is shaped like a right triangle. FLAGS Write a coordinate proof to prove this flag is shaped like an isosceles triangle. The length is 16 inches and the height is 10 inches. C Proof: Vertex A is at the origin and B is at (0, 10). The x-coordinate of C is 16. The y-coordinate is halfway between 0 and 10 or 5. So, the coordinates of C are (16, 5). Determine the lengths of CA and CB. Since each leg is the same length, ABC is isosceles. The flag is shaped like an isosceles triangle. Explore online information about the information introduced in this chapter. Click on the Connect button to launch your browser and go to the Glencoe Geometry Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to www.geometryonline.com/extra_examples. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. End of Custom Shows WARNING! Do Not Remove This slide is intentionally blank and is set to auto-advance to end custom shows and return to the main presentation.