Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

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3 Name: lass: _ ate: _ I: SSS Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Given the lengths marked on the figure and that bisects E, use SSS to explain d. The triangles are not congruent. 2. The figure shows part of the roof structure of a house. Use SS to explain why RTU. omplete the explanation. It is given that [1]. Since RTS and RTU are right angles, [2] by the Right ngle ongruence Theorem. y the Reflexive Property of ongruence, [3]. Therefore, RTU by SS. a. [1] RT [2] URT [3] UT b. [1] UT [2] URT [3] UT c. [1] UT [2] RTU [3] RT d. [1] UT [2] RTU [3] SU 1

4 Name: I: 3. What additional information do you need to by the SS Postulate? 4. etermine if you can use S to E. Explain. is E because both are right angles. No other congruence relationships can be determined, so S cannot be applied. is E because both are right angles. y the djacent ngles E. E by S. is E because both are right angles. y the Vertical ngles E. E by S. is E because both are right angles. y the Vertical ngles E. E by SS. 2

5 Name: I: 5. etermine if you can use the HL ongruence Theorem to If not, tell what else you need to know. a. Yes. b. No. You do not know that and are right angles. c. No. You do not know d. No. You do not know that Ä. 6. For these triangles, select the triangle congruence statement and the postulate or theorem that supports it. JLK, HL JLK, SS JKL, HL JKL, SS 3

6 Name: I: 7. pilot uses triangles to find the angle of elevation from the ground to her plane. How can she find m? a. O by SS by PT, so m = 40 by substitution. b. O by PT by SS, so m = 40 by substitution. c. O by S by PT, so m = 40 by substitution. d. O by PT by S, so m = 40 by substitution. 8. Find the value of x. a. x = 6 c. x = 2 b. x = 4 d. x = 8 9. Find m Q. a. m Q = 30 º c. m Q = 70 º b. m Q = 60 º d. m Q = 75 º 4

7 Name: I: 10. Find. a. = 10 b. = 12 c. = 14 d. Not enough information. n equiangular triangle is not necessarily equilateral. 11. Find the measure of each numbered angle. a. m 1 = 54, m 2 = 117, m 3 = 63 b. m 1 = 117, m 2 = 63, m 3 = 63 c. m 1 = 54, m 2 = 63, m 3 = 63 d. m 1 = 54, m 2 = 63, m 3 = 117 5

8 I: SSS nswer Section MULTIPLE HOIE 1. NS: It is given E, and bisects E. y the definition of segment ll three pairs of corresponding sides of the triangles are congruent. E by SSS. orrect! Use the fact that segment bisects segment E. The corresponding sides need to belong to different triangles. Use the fact that segment bisects segment E. The corresponding sides of the triangles are congruent. Use the fact that segment bisects segment E. PTS: 1 IF: asic REF: Page 242 OJ: Using SSS to Prove Triangle ongruence NT: a TOP: 4-4 Triangle ongruence: SSS and SS 2. NS: It is given that UT. Since RTS and RTU are right angles, RTU by the Right ngle ongruence Theorem. y the Reflexive Property of ongruence, RT. Therefore, RTU by SS. heck the figure to see what is given. ngle SRT and angle URT are not right angles. orrect! Segment SU being congruent to itself does not help in proving the triangles congruent. PTS: 1 IF: verage REF: Page 243 OJ: pplication NT: a TOP: 4-4 Triangle ongruence: SSS and SS 1

9 I: 3. NS: The SS Postulate is used when two sides and an included angle of one triangle are congruent to the corresponding sides and included angle of a second triangle. From the From the by the Reflexive Property of ongruence. You have two pair of congruent sides, so you need information about the included angles. Use these pairs of sides to determine the included angles. The angle between sides and is. The angle between sides and is. You need to to by the SS Postulate. This information is needed to use the SSS Postulate. orrect! You need the included angle between the two sides. This information is already given. Find information that you need that is not given or true in the figure. PTS: 1 IF: dvanced NT: a TOP: 4-4 Triangle ongruence: SSS and SS 4. is E because both are right angles. y the Vertical ngles E. E by S. Look for vertical angles. djacent angles are angles in a plane that have their vertex and one side in common but have no interior points in common. ngle and angle E are not adjacent angles. orrect! Use S, not SS, to prove the triangles congruent. PTS: 1 IF: asic REF: Page 253 OJ: pplying S ongruence NT: e TOP: 4-5 Triangle ongruence: S S and HL 2

10 I: 5. NS: Ä is given. In addition, by the Reflexive Property of Since Ä and ^P, by the Perpendicular Transversal Theorem ^P. y the definition of right angle, is a right angle. Similarly, is a right angle. by the HL ongruence Theorem. orrect! Since line segment is parallel to line segment, what does the Perpendicular Transversal Theorem tell you about line segment and line segment P? What do you know about the other pair of legs of the right triangles and? What do you know about line segments and? PTS: 1 IF: verage REF: Page 255 OJ: pplying HL ongruence NT: e TOP: 4-5 Triangle ongruence: S S and HL 6. NS: ecause and KJL are right angles, and JKL are right triangles. You are given a pair of congruent JL and a pair of congruent LK. So a hypotenuse and a leg of are congruent to the corresponding hypotenuse and leg of JKL by HL. Segment is congruent to segment JL. Make sure the triangle vertices correspond accordingly. orrect! Segment is congruent to segment JL. Make sure the triangle vertices correspond accordingly. For SS, the angle is included between the sides. For SS, the angle is included between the sides. PTS: 1 IF: dvanced NT: a TOP: 4-5 Triangle ongruence: S S and HL 3

11 I: 7. NS: From the figure, O,and O. O by the Vertical ngles Theorem. Therefore, O by SS by PT. m = 40 by substitution. orrect! First, show that the triangles are congruent. Then, show that their corresponding parts are congruent. First, show that the triangles are congruent. Then, show that their corresponding parts are congruent. First, show that the triangles are congruent. Then, show that their corresponding parts are congruent. PTS: 1 IF: verage REF: Page 260 OJ: pplication NT: e TOP: 4-6 Triangle ongruence: PT 8. NS: The triangles can be proved congruent by the SS Postulate. y PT, 3x - 5 = 2x + 1. Solve the equation for x. 3x - 5 = 2x + 1 3x = 2x + 6 x = 6 orrect! When solving, you can either add 5 or subtract 1 from each side. Remember to combine the like terms when solving. These two triangles have SS congruence, so the two expressions are equal by PT. PTS: 1 IF: dvanced NT: e TOP: 4-6 Triangle ongruence: PT 4

12 I: 9. NS: m Q = m R = ( 2x + 15) Isosceles Triangle Theorem m P + m Q + m R = 180 Triangle Sum Theorem x + ( 2x + 15) + ( 2x + 15) = 180 Substitute x for m P and substitute 2x + 15 for m Q and m R. 5x = 150 Simplify and subtract 30 from both sides. x = 30 ivide both sides by 5. Thus m Q = ( 2x + 15) = [2( 30) + 15] = 75. This is x. The measure of angle Q is 2x y the Isosceles Triangle Theorem, the measure of angle Q equals the measure of angle R. Use the Triangle Sum Theorem and solve for x. y the Isosceles Triangle Theorem, the measure of angle Q equals the measure of angle R. Use the Triangle Sum Theorem and solve for x. orrect! PTS: 1 IF: verage REF: Page 274 OJ: Finding the Measure of an ngle NT: f TOP: 4-8 Isosceles and Equilateral Triangles 10. NS: is equilateral. Equiangular triangles are equilateral. 2s - 10 = s + 2 efinition of equilateral triangle. s = 12 Subtract s and add 10 to both sides of the equation. = 2s - 10 = 2( 12) - 10 Substitute 12 for s in the equation for. = 14 Simplify. = = 14 efinition of equilateral triangle. Substitute 14 for. Equiangular triangles are equilateral. Use = to solve for s, and then use = or = to find. This is s. Substitute s in the original equation to find. orrect! y a corollary to the Isosceles Triangle Theorem, equiangular triangles are equilateral. Use = to solve for s, and then use = or = to find. PTS: 1 IF: asic REF: Page 275 OJ: Using Properties of Equilateral Triangles TOP: 4-8 Isosceles and Equilateral Triangles NT: f 5

13 I: 11. NS: Step 1: 2 is supplementary to the angle that is m 2 = 180. So m 2 = 63. Step 2: y the lternate Interior ngles Theorem, 3. So m 2 = m 3 = 63. Step 3: y the Isosceles Triangle Theorem, 2 and the angle opposite the other side of the isosceles triangle are congruent. Let 4 be that unknown angle. Then, 4 and m 2 = m 4 = 63. m 1 + m 2 + m 4 = 180 by the Triangle Sum Theorem. m = 180. So m 1 = 54. ngle 2 is supplementary to the angle that measures 117 degrees. To find the measure of angle 1, use the Isosceles Triangle Theorem. orrect! y the lternate Interior ngles Theorem, angle 2 is congruent to angle 3. PTS: 1 IF: dvanced NT: e TOP: 4-8 Isosceles and Equilateral Triangles KEY: multi-step 6