Triangle LMO is congruent to triangle LNP. Given that angle PMN is congruent to angle ONM and angle MPN is congruent to angle NOM. Triangle LMN is isosceles so LN is congruent to LM. Triangle MPN is congruent to triangle NOM by the SAA congruence criterion. Therefore, triangle LMO is congruent to triangle LNP by the SAS congruence criterion. The measure of angle RSU is 90 degrees. Given that line RS is congruent to line TS and line RU is congruent to line TU. Triangle RUS is congruent to triangle TUS by the SSS congruence criterion. Therefore, angle R

2. 6.7 PROBLEM 14 PG. 249
Given: Angle PMN is congruent to angle ONM; Angle MPN is congruent to angle NOM
Prove: Triangle LMO is congruent to triangle LNP

3. STEP 1
AN G L E PMN I S CON G RUEN T TO
AN G L E ON M; AN G L E MPN IS
CON G RUEN T TO AN G L E N OM GIVEN

4. STEP 2
TRIANGLE LMN IS AN
ISOSCELES TRIANGLE; DEFINITION OF
LINE LN IS CONGRUENT ISOSCELES TRIANGLE
TO LINE LM

5. STEP 3
LINE MN IS CONGRUENT REFLEXIVE PROPERTY OF
TO LINE MN CONGRUENT SEGMENTS

6. STEP 4
TRIANGLE MPN IS SAA TRIANGLE
CONGRUENT TO TRIANGLE
NOM PO ST U LAT E

7. STEP 5
LINE PN IS CONGRUENT TO DEFINITION OF
LINE OM, ANGLE 2 IS
CONGRUENT TO ANGLE 3 CONGRUENT TRIANGLES

8. STEP 6
ANGLE 1 IS CONGRUENT ADJACENT ANGLE
TO ANGLE 4 PORTION THEOREM

9. STEP 7
TRIANGLE LMO IS
SAS TRIANGLE
CONGRUENT TO TRIANGLE
PO ST U LAT E
NOM

10. 6.8 PROBLEM 18 PG. 255
Given: Line RS is congruent to line TS; Line RU is congruent to line TU
Prove: The measure of angle RSU is 90 degrees

11. STEP 1
LINE RS IS CONGRUENT TO
LINE TS; LINE RU IS
CONGRUENT TO LINE TU GIVEN

12. STEP 2
LINE US IS CONGRUENT REFLEXIVE PROPERTY OF
TO LINE US CONGRUENT SEGMENTS

13. STEP 3
TRIANGLE RUS IS SSS TRIANGLE
CONGRUENT TO TRIANGLE
TUS PO ST U LAT E

14. STEP 4
ANGLE RSU IS CONGRUENT SSS TRIANGLE
TO ANGLE TSU PO ST U LAT E

15. STEP 5
ANGLE RSU AND ANGLE LINEAR PAIRS ARE
TSU ARE SUPPLEMENTARY SUPPLEMENTARY

16. STEP 6
ANGLE RSU AND ANGLE CONGRUENT SUPPLEMENTS
TSU ARE RIGHT ANGLES ARE RIGHT ANGLES

17. STEP 7
THE MEASURE OF ANGLE DEFINITION OF RIGHT
RSU IS 90 DEGREES ANGLE