Question

1. a supplement of an angle is twice as large as the angle. find the measure of the angle.
2. the difference between the measures of two supplementary angles is 42. find both angles.

do each proof below.

1. given: $overline{pt}paralleloverline{wv}$, $overline{rq}paralleloverline{su}$ prove: $angle1congangle2$
2. given: $overline{cp}paralleloverline{ql}$, $angle qcongangle l$ prove: $angle ccongangle p$

Explanation:

20.

Step1: Define the angle and its supplement

Let the angle be $x$. Its supplement is $180 - x$.

Step2: Set up the equation

We know that $180 - x=2x$.

Step3: Solve the equation

Add $x$ to both sides: $180=2x + x$. So, $180 = 3x$. Then $x=\frac{180}{3}=60$.

Step1: Let the two supplementary angles

Let one angle be $x$ and the other be $y$. We know that $x + y=180$ and $|x - y| = 42$.

Step2: Case 1: $x-y = 42$

From $x + y=180$ and $x - y=42$, add the two - equations: $(x + y)+(x - y)=180 + 42$. This gives $2x=222$, so $x = 111$. Substitute $x = 111$ into $x + y=180$, we get $111+y=180$, then $y=180 - 111 = 69$.

Step3: Case 2: $y - x=42$

From $x + y=180$ and $y - x=42$, add the two - equations: $(x + y)+(y - x)=180+42$. This gives $2y=222$, so $y = 111$. Substitute $y = 111$ into $x + y=180$, we get $x=180 - 111 = 69$.

Step1: Use the property of parallel lines

Since $\overline{PT}\parallel\overline{WV}$, $\angle 1$ and $\angle PMR$ are corresponding angles, so $\angle 1=\angle PMR$.

Step2: Use the property of parallel lines again

Since $\overline{RQ}\parallel\overline{SU}$, $\angle PMR$ and $\angle 2$ are corresponding angles, so $\angle PMR=\angle 2$.

Step3: Transitive property

By the transitive property of equality, if $\angle 1=\angle PMR$ and $\angle PMR=\angle 2$, then $\angle 1\cong\angle 2$.

Answer:

$60^{\circ}$

21.