12 Write a two-column proof.

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1 Name: Period: Date: ID: A Chapter 4 Review -- GH 1 Find each measure. m 1, m 2, m 3 9 Triangles ABC and AFD are vertical congruent equilateral triangles. Find x and y. 2 Determine whether PQR STU given the coordinates of the vertices. Explain. PÊ Á 0,3 ˆ,Q Ê Á 0, 1 ˆ,R Ê Á 2, 1ˆ SÊ Á 1,2 ˆ,T Ê Á 1, 2 ˆ,U Ê Á 1, 2ˆ Refer to the figure below for problems What is m RAM? 4 What is m AMX? 5 What is m MAX? 10 Triangles MNP and OMN are congruent equilateral triangles. Find x and y. 11 The measures of the angles of ABC are 5x+25,9x 3,and8x+4. a. Draw a figure to illustrate ABC. b. Find the measure of each angle of ABC. Explain. c. Prove (algebraically) that ABC is an equilateral triangle. Do not write a proof. 12 Write a two-column proof. Given: W is the midpoint of XY and VZ; XV YZ. Prove: XVW YZW 6 If m FXA=96, what is m XFM? 7 What is m ARM? 8 If m FMX=23 what is m FXA? 1

2 Show all work for full credit. ID: A 13 Write a two column proof for the problem. Given: ADB CDB;DBbisects ABC. Prove: ABD CBD 16 Given m NOP=m NPO; NO=2n+4; NP=8n 3; OP=9; find NO. 14 Write a two column proof. Given: BD bisects ABC, AB BC Prove:AD CD 17 Use the Distance Formula to find the length of each side of WXY. Classify the triangle by its sides. WÊ Á0,0ˆ, XÊ Á a,b ˆ, Y Ê Á 2a,0 ˆ 18 What other congruence must be given for ABC EDC by the SAS Congruence Postulate? 15 Write a two column proof. Given: BAC DAC, DCA BCA Prove:BC DC 19 What is the measure of each base angle of an isosceles triangle if its vertex angle measures 40 degrees and its 2 congruent sides measure 25 units? 20 In ABC, if AB BC and m A= 39, then m C=. 21 Can you use the SAS Congruence Postulate to prove that the two triangles are congruent? Explain your reasoning. 2

3 Show all work for full credit. ID: A 22 Position and label the triangle on the coordinate plane. right isosceles ABC with congruent sides AB and AC a units long 24 Position and label the triangle on the coordinate plane. right ABC with hypotenuse AB, leg AC 4a units long, and leg BC one-fourth the other leg F. F. G. G. H. H. I. 23 Use information in the figure below to find m D. I. 25 Write a two column proof. Given: ED EC;BD BC;ED BC Prove: CED DBC 3

Show all work for full credit. ID: A 26 Triangle RST is isosceles, with RS STand SRT STR.(The figure may not be drawn to scale.) 32 MN joins the midpoint of AB and the midpoint of AC in ABC.

4 Show all work for full credit. ID: A 26 Triangle RST is isosceles, with RS STand SRT STR.(The figure may not be drawn to scale.) 32 MN joins the midpoint of AB and the midpoint of AC in ABC. Find the coordinates of M and N. Write an equation that can be solved to find the value of x. Solve the equation in and use the answer to find the measure, in degrees, of RST. 33 Find each measure. m 1, m 2, m 3 27 If TGS KEL, which angle in KEL corresponds to T? 28 Which triangles are congruent in the figure? Write the congruency statment. 34 Triangle FJH is an equilateral triangle. Find x and y. 29 Find m Find the value of x. 31 Position and label equilateral KLM with side lengths 3a units long on the coordinate plane. 4

5 ID: A Chapter 4 Review -- GH Answer Section 1 m 1=141,m 2=84, m 3=139 2 Yes; Each side of triangle PQR is the same length as the corresponding side of triangle STU x=7,y=27 10 x=8,y=18 11 Sample: a. b. From the Angle Sum Theorem we know that ( 8x+4)+ ( 9x 3)+ ( 5x+25) =180 22x+26=180 22x=154 x=7 Substituting x back into the angles we get m A=5(7)+25=60 m B=9(7) 3=60 m C=8(7)+4=60 c. Since all of the angles are congruent, the triangle is equiangular. An equiangular triangle is also equilateral, so ABC is equilateral. 1

6 ID: A 12 Sample: Given: W is the midpoint of XY and VZ; XV YZ. Prove: XVW YZW Proof: Statements Reasons 1. W is the midpoint of XY. 1. Given 2. XW YW 2. Midpoint Theorem 3. W is the midpoint of VZ. 3. Given 4. VW ZW 4. Midpoint Theorem 5. XV YZ 5. Given 6. XVW YZW 6. SSS Postulate 13 Given: ADB CDB;DBbisects ABC Prove: ABD CBD Statements Reasons 1. ADB CDB;DBbisects ABC 1. Given 2. DB DB 2. Reflexive Property 3. ABD CBD 3. Def. of Angle Bisector 4. ABD CBD 4. ASA 14 ABD CBD 15 Statements Reasons 1. BAC DAC, DCA BCA 1.Given 2. AC AC 2. Reflexive properpty of congruence 3. ABC ADC 3. ASA Congruence Postulate 4. BC DC 4. Corresponding parts of congruent s are congruent 16 NO =

7 ID: A 17 WY=2a WX= ( a 0) 2 + ( b 0) 2 = ( a) 2 + ( b) 2 or a 2 +b 2 XY= ( 2a a) 2 + ( 0 b) 2 = ( a) 2 + ( b) 2 or a 2 +b 2 Isosceles 18 AC CE No. In order to use the SAS Congruence Postulate, the sides that form the congruent angles must be congruent. In the diagram, the right angles are congruent, but only 1 pair of sides that forms the right angle are congruent. 22 G I Statements Reasons 1. ED EC 1.Given 2. CEDisart 2.Defof lines BD BC 3.Given 4. DBCisart 4.Defof lines 5. ED BC 5.Given 6. DC CD 6.Reflexive 7. CED DBC 7.HLCongruenceTheorem 26 ( 4x 27)+5x=180. m SRT = K 28 HMN NGH (sample answer) x = 6 3

8 ID: A 31 Ê 32 M a 2,0 ˆ Á, N Ê 0,b ˆ Á 2 33 m 1=77,m 2=36, m 3=30 34 x=7,y=16 4

Properties of Isosceles and Equilateral Triangles

Properties of Isosceles and Equilateral Triangles In an isosceles triangle, the sides and the angles of the triangle are classified by their position in relation to the triangle s congruent sides. Leg