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October 26th, 2016, 03:24 AM   #1
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Proving angle Relationships help

Could anyone please help me with these questions?


Last edited by skipjack; November 9th, 2016 at 05:39 AM.
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November 9th, 2016, 06:38 AM   #2
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The answers below may not be word perfect.

I've done the arithmetic for you, but haven't named every theorem used. Some results may follow from definitions rather than theorems.

A "straight angle" is (measures) 180°.

1. The two angles combine to form a straight angle, and you are told the second is 57°, so the first is 123°. Not sure of the best theorem to cite here - possibly the straight angles theorem.

2. The right angle shown is 90°, so the same principle as that used in (1) tells you that angle 6 is 180° - 90° - 22° = 68°.

3. By the vertical angle theorem, the numbered angles are equal, so each measures 38°.

4. 4x + 11 + 3x + 1 = 180, so 7x = 168, which implies x = 24. Hence angles 13 and 14 are respectively 107° and 73°.

5. By definition, two angles are complementary if their sum is 90°. As for (2), angle 9 = angle 7 = 180° - 90° - 41° = 49°. Angle 10 = 180° - 49° - 41° - 49° = 41°.

6. By the vertical angle theorem, 4x - 26 = 3x + 4, so x = 30. Hence angle 2 and angle 3 are each 94°.

7.
a. $\small\angle$QPS $\cong$ $\small\angle$TPR | given
b. Congruent angles have equal measure by definition.
c. Angle addition
d. m$\small\angle$QPR + m$\small\angle$RPS
$\ \ \ $ m$\small\angle$TPS + m$\small\angle$RPS
e. m$\small\angle$QPR = m$\small\angle$TPS | subtraction property
f. $\small\angle$QPR $\cong$ $\small\angle$TPS | definition of conguent angles
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