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Contents Lesson 4-1Classifying Triangles Lesson 4-2Angles of Triangles Lesson 4-3Congruent Triangles Lesson 4-4Proving Congruence–SSS, SAS Lesson 4-5Proving Congruence–ASA, AAS Lesson 4-6Isosceles Triangles Lesson 4-7Triangles and Coordinate Proof

Lesson 1 Contents Example 1Classify Triangles by Angles Example 2Classify Triangles by Sides Example 3Find Missing Values Example 4Use the Distance Formula

Example 1-1a ARCHITECTURE The triangular truss below is modeled for steel construction. Classify  JMN,  JKO, and  OLN as acute, equiangular, obtuse, or right.

Example 1-1a 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.

Example 1-1b 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.

Example 1-2a Answer:  UTX and  UVX are isosceles. Identify the isosceles triangles in the figure if Isosceles triangles have at least two sides congruent.

Example 1-2b Identify the scalene triangles in the figure if Answer:  VYX,  ZTX,  VZU,  YTU,  VWX,  ZUX, and  YXU are scalene. Scalene triangles have no congruent sides.

Example 1-2c Identify the indicated triangles in the figure. a. isosceles triangles b. scalene triangles Answer:  ABC,  EBC,  DEB,  DCE,  ADC,  ABD Answer:  ADE,  ABE

Example 1-3a 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.

Example 1-3b Next, substitute to find the length of each side. Answer: For  KLM, and the measure of each side is 7.

Example 1-3c Answer: ALGEBRA Find d and the measure of each side of equilateral triangle if and

Example 1-4a COORDINATE GEOMETRY Find the measures of the sides of  RST. Classify the triangle by sides.

Example 1-4b Answer: ; since all 3 sides have different lengths,  RST is scalene. Use the distance formula to find the lengths of each side.

Example 1-4c Find the measures of the sides of  ABC. Classify the triangle by sides. Answer: ; since all 3 sides have different lengths,  ABC is scalene.

End of Lesson 1

Lesson 2 Contents Example 1Interior Angles Example 2Exterior Angles Example 3Right Angles

Example 2-1a 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.

Example 2-1a Answer: Angle Sum Theorem Simplify. Subtract 142 from each side.

Example 2-1b Find the missing angle measures. Answer:

Example 2-2a Find the measure of each numbered angle in the figure. Exterior Angle Theorem Simplify. Substitution Subtract 70 from each side. If 2  s form a linear pair, they are supplementary.

Example 2-2a Exterior Angle Theorem Subtract 64 from each side. Substitution Subtract 78 from each side. If 2  s form a linear pair, they are supplementary. Substitution Simplify.

Example 2-2a Subtract 143 from each side. Angle Sum Theorem Substitution Simplify. Answer:

Example 2-2b Find the measure of each numbered angle in the figure. Answer:

Example 2-3a Corollary 4.1 Substitution Subtract 20 from each side. Answer: GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20.

Answer: Example 2-3b The piece of quilt fabric is in the shape of a right triangle. Find if is 32.

End of Lesson 2

Lesson 3 Contents Example 1Corresponding Congruent Parts Example 2Transformations in the Coordinate Plane

Example 3-1a 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 corresponding congruent angles and sides of  HIJ and  LIK.

Example 3-1b Name the congruent triangles. ARCHITECTURE A tower roof is composed of congruent triangles all converging toward a point at the top. Answer:  HIJ  LIK

Example 3-1c Answer: The support beams on the fence form congruent triangles. b. Name the congruent triangles. a. Name the corresponding congruent angles and sides of  ABC and  DEF. Answer:  ABC  DEF

Example 3-2a 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. Example 3-2b

Use the Distance Formula to find the length of each side of the triangles. Example 3-2b

Use the Distance Formula to find the length of each side of the triangles. Example 3-2b

Use a protractor to measure the angles of the triangles. You will find that the measures are the same. Example 3-2c Answer: The lengths of the corresponding sides of two triangles are equal. Therefore, by the definition of congruence, TempCopy In conclusion, because,

Example 3-2d 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.

Example 3-2f 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). Answer: Use a protractor to verify that corresponding angles are congruent. a. Verify that  ABC  ABC.

Example 3-2g Answer: turn b. Name the congruence transformation for  ABC and  ABC.

End of Lesson 3

Lesson 4 Contents Example 1Use SSS in Proofs Example 2SSS on the Coordinate Plane Example 3Use SAS in Proofs Example 4Identify Congruent Triangles

Example 4-1a 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

Example 4-1b Given: G is the midpoint of both Prove: 1. Given1. Proof: ReasonsStatements 3. SSS 3.  FEG  HIG  FEG  HIG 2. Midpoint Theorem2.

Example 4-1b 3. SSS 1. Given 2. Reflexive Proof: ReasonsStatements  ABC  GBC Write a two-column proof to prove that  ABC  GBC if

Example 4-2a Use the Distance Formula to show that the corresponding sides are congruent. 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.

Example 4-2b Answer: By definition of congruent segments, all corresponding segments are congruent. Therefore,  WDV  MLP by SSS.

Example 4-2c Answer: By definition of congruent segments, all corresponding segments are congruent. Therefore,  ABC  DEF 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.

Example 4-3a Write a flow proof. Given: Prove:  QRT  STR

Example 4-3b Answer:

Example 4-3c Write a flow proof. Given:. Prove:  ABC  ADC

Example 4-3d Proof:

Example 4-4a 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. Answer: SAS 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.

Example 4-4b 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. Answer: SSS 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.

Example 4-4c 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. a.

Example 4-4d Answer: not possible b.

End of Lesson 4

Lesson 5 Contents Example 1Use ASA in Proofs Example 2Use AAS in Proofs Example 3Determine if Triangles Are Congruent

Example 5-1a 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: L is the midpoint of Prove:  WRL  EDL

Example 5-1b Proof: because alternate interior angles are congruent. because alternate interior angles are congruent. by Reflexive Property.  ABD  CDB by ASA. Write a paragraph proof. Given: Prove:  ABD  CDB

Example 5-2a Write a flow proof. Given: Prove:

Example 5-2b Proof:

Example 5-2c Proof: Given: Prove: Write a flow proof.

Example 5-3a 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.

Example 5-3b 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. Answer:  ABC  EDF by SSS 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

Example 5-3c Answer:  ABE  CBD 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.

End of Lesson 5

Lesson 6 Contents Example 1Proof of Theorem Example 2Find the Measure of a Missing Angle Example 3Congruent Segments and Angles Example 4Use Properties of Equilateral Triangles

Example 6-1a Write a two-column proof. Given: Prove:

Example 6-1b Proof: ReasonsStatements 3. Def. of isosceles  3.  ABC and  BCD are isosceles 1. Given Substitution 5.5. Given Isosceles  Theorem 2. Def. of segments2.

Example 6-1c Write a two-column proof. Given:. Prove:

Example 6-1d Proof: ReasonsStatements 1. Given 3. Isosceles  Theorem 2. Def. of isosceles triangles  ADB is isosceles Given 5. Def. of midpoint 6. SAS 7.7. CPCTC 6.  ABC  ADC

Example 6-2a Multiple-Choice Test Item If and what is the measure of Read the Test Item  CDE is isosceles with base Likewise,  CBA is isosceles with A B C D. 75

Example 6-2b Solve the Test Item Angle Sum Theorem Substitution Add. Subtract 120 from each side. Divide each side by 2. Step 1 The base angles of  CDE are congruent. Let

Example 6-2c Def. of vertical angles Substitution Add. Step 2 are vertical angles so they have equal measures. Step 3 The base angles of  CBA are congruent. Angle Sum Theorem Substitution Subtract 30 from each side. Divide each side by 2.

Example 6-2d Answer: D

Example 6-2e Answer: A Multiple-Choice Test Item If and what is the measure of A. 25 B. 35 C. 50 D. 130

Example 6-3a Answer: Name two congruent angles.

Example 6-3b Answer: Name two congruent segments. By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,

Example 6-3c a. Name two congruent angles. Answer: b. Name two congruent segments.

Example 6-4a Since the angle was bisected, Each angle of an equilateral triangle measures 60°.  EFG is equilateral, and bisects bisects Find and

Example 6-4b Answer: Add. Exterior Angle Theorem Substitution is an exterior angle of  EGJ.

Example 6-4c Subtract 75 from each side. Linear pairs are supplementary. Substitution Answer: 105  EFG is equilateral, and bisects bisects Find

a. Find x. Example 6-4d b. Answer: 90 Answer: 30  ABC is an equilateral triangle. bisects

End of Lesson 6

Lesson 7 Contents Example 1Position and Label a Triangle Example 2Find the Missing Coordinates Example 3Coordinate Proof Example 4Classify Triangles

Example 7-1a Use the origin as vertex X of the triangle. Place the base of the triangle along the positive x-axis. Position and label right triangle XYZ with leg d units long on the coordinate plane. X (0, 0) Z (d, 0) Position the triangle in the first quadrant. Since Z is on the x-axis, its y-coordinate is 0. Its x-coordinate is d because the base is d units long.

Since triangle XYZ is a right triangle the x-coordinate of Y is 0. We cannot determine the y-coordinate so call it b. Example 7-1b Answer: X (0, 0) Z (d, 0) Y (0, b)

Example 7-1c Answer: Position and label equilateral triangle ABC with side w units long on the coordinate plane.

Example 7-2a Name the missing coordinates of isosceles right triangle QRS. Answer: Q(0, 0); S(c, c) 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).

Example 7-2b Answer: C(0, 0); A(0, d) Name the missing coordinates of isosceles right  ABC.

Example 7-3a 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.

Example 7-3b Prove: 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.

Example 7-3c Proof: By the Midpoint Formula, the coordinates of W, the midpoint of, is The slope of or undefined. The slope of is therefore,.

Example 7-3d 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.

Example 7-3e Proof: The coordinates of the midpoint D are The slope of is or 1. The slope of or –1, therefore.

Example 7-4a 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.

Example 7-4b 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

Example 7-4c 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.

End of Lesson 7

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