Question

in exercises 1 - 4, identify the pair(s) of congruent angles in the figures. explain how you know they are congruent.
1.
2.
3.

1. ∠abc is supplementary to ∠cbd. ∠cbd is supplementary to ∠def.

in exercises 5 - 8, use the diagram and the given angle measure to find the other three measures.

1. m∠1 = 143°
2. m∠3 = 159°
3. m∠2 = 34°
4. m∠4 = 29°

in exercises 9 - 12, find the value of each variable. (see example 4.)
9.
10.

Explanation:

Exercise 1:

Given that $\angle MSN = 50^{\circ}$ and $\angle QSR=50^{\circ}$, by the definition of congruent angles (angles with equal measures), $\angle MSN\cong\angle QSR$.

Exercise 2:

In right - angle $\angle FGH = 90^{\circ}$, $\angle JKM$ has a measure of $45^{\circ}+45^{\circ}=90^{\circ}$. Since $\angle FGH$ and $\angle JKM$ have equal measures ($90^{\circ}$), $\angle FGH\cong\angle JKM$.

Exercise 3:

Vertical angles are congruent. $\angle GMH$ and $\angle JML$ are vertical angles, so $\angle GMH\cong\angle JML$. Also, $\angle GML$ and $\angle JMH$ are vertical angles, so $\angle GML\cong\angle JMH$.

Exercise 4:

If $\angle ABC$ is supplementary to $\angle CBD$ and $\angle CBD$ is supplementary to $\angle DEF$, then by the transitive property of supplementary angles, $\angle ABC\cong\angle DEF$.

Exercise 5:

If $m\angle1 = 143^{\circ}$, then $\angle1$ and $\angle3$ are vertical angles, so $m\angle3=143^{\circ}$. $\angle1$ and $\angle2$ are supplementary, so $m\angle2 = 180^{\circ}-143^{\circ}=37^{\circ}$. Since $\angle2$ and $\angle4$ are vertical angles, $m\angle4 = 37^{\circ}$.

Exercise 6:

If $m\angle3 = 159^{\circ}$, then $m\angle1 = 159^{\circ}$ (vertical angles). $m\angle2=180^{\circ}-159^{\circ}=21^{\circ}$, and $m\angle4 = 21^{\circ}$ (vertical angles).

Exercise 7:

If $m\angle2 = 34^{\circ}$, then $m\angle4 = 34^{\circ}$ (vertical angles). $m\angle1=180^{\circ}-34^{\circ}=146^{\circ}$, and $m\angle3 = 146^{\circ}$ (vertical angles).

Exercise 8:

If $m\angle4 = 29^{\circ}$, then $m\angle2 = 29^{\circ}$ (vertical angles). $m\angle1=180^{\circ}-29^{\circ}=151^{\circ}$, and $m\angle3 = 151^{\circ}$ (vertical angles).

Exercise 9:

Since vertical angles are equal, we set up the equation $16y=18y - 18$.

Step1: Rearrange the equation

Subtract $16y$ from both sides: $0=18y-16y - 18$, which simplifies to $0 = 2y-18$.

Step2: Solve for $y$

Add 18 to both sides: $18 = 2y$. Then divide both sides by 2, so $y = 9$.

Exercise 10:

Since vertical angles are equal, we set up the equation $8x + 7=9x-4$.

Step1: Rearrange the equation

Subtract $8x$ from both sides: $7=x - 4$.

Step2: Solve for $x$

Add 4 to both sides: $x=11$.

Answer:

Exercise 1: $\angle MSN\cong\angle QSR$
Exercise 2: $\angle FGH\cong\angle JKM$
Exercise 3: $\angle GMH\cong\angle JML$, $\angle GML\cong\angle JMH$
Exercise 4: $\angle ABC\cong\angle DEF$
Exercise 5: $m\angle2 = 37^{\circ}$, $m\angle3 = 143^{\circ}$, $m\angle4 = 37^{\circ}$
Exercise 6: $m\angle1 = 159^{\circ}$, $m\angle2 = 21^{\circ}$, $m\angle4 = 21^{\circ}$
Exercise 7: $m\angle1 = 146^{\circ}$, $m\angle3 = 146^{\circ}$, $m\angle4 = 34^{\circ}$
Exercise 8: $m\angle1 = 151^{\circ}$, $m\angle2 = 29^{\circ}$, $m\angle3 = 151^{\circ}$
Exercise 9: $y = 9$
Exercise 10: $x = 11$