Question

using angle relationships to find angle measures
directions: find the missing measures in each figure. keep the angle relationships in mind.
1.
2.
3.
4.
5.

1. ∠1 and ∠2 are vertical angles. if the measure of ∠2 is 105°, find the measure of ∠1.
2. ∠a and ∠b are complementary angles. if the measure of ∠a is 42°, find the measure of ∠b.
3. ∠p and ∠q are supplementary angles. if the measure of ∠q is 64°, find the measure of ∠p.
4. ∠1 and ∠2 form a linear pair. if the measure of ∠1 is 113°, find the measure of ∠2.

using algebra

1. if ( mangle pqt=(3x + 47)^{circ} ) and ( mangle sqr=(6x - 25)^{circ} ), find the measure of ( angle sqr ).
2. if ( overrightarrow{ab}perpoverrightarrow{cd} ), ( mangle dce=(7x + 2)^{circ} ) and ( mangle ecb=(x + 8)^{circ} ), find the measure of ( angle dce ).
3. if ( mangle knm=(8x - 5)^{circ} ) and ( mangle mnj=(4x - 19)^{circ} ), find the measure of ( angle knm ).

Explanation:

1.

Step1: Vertical angles are equal

Since \(x\) and \(112^{\circ}\) are vertical angles, \(x = 112^{\circ}\)

2.

Step1: Complementary angles sum to \(90^{\circ}\)

\(x+68 = 90\)

Step2: Solve for \(x\)

\(x=90 - 68=22^{\circ}\)

3.

Step1: Supplementary angles sum to \(180^{\circ}\)

\(x + 124=180\)

Step2: Solve for \(x\)

\(x=180 - 124 = 56^{\circ}\)

4.

Step1: Vertical angles

\(x = 43^{\circ}\) (vertical angles with \(43^{\circ}\))

Step2: Supplementary angles

\(y+43 = 180\), so \(y = 180 - 43=137^{\circ}\)

Step3: Vertical angles

\(z=y = 137^{\circ}\) (vertical angles with \(y\))

5.

Step1: Vertical angles

\(y = 72^{\circ}\) (vertical angles with \(72^{\circ}\))

Step2: \(x + 72=180\) (linear pair)

\(x=180 - 72 = 108^{\circ}\)

Step3: \(z+72 = 90\) (complementary as \(x\) and \(z\) with right - angle relation)

\(z=90 - 72=18^{\circ}\)

6.

Step1: Vertical angles are equal

Since \(\angle1\) and \(\angle2\) are vertical angles, \(m\angle1=m\angle2 = 105^{\circ}\)

7.

Step1: Complementary angles sum to \(90^{\circ}\)

\(m\angle B=90 - 42=48^{\circ}\)

8.

Step1: Supplementary angles sum to \(180^{\circ}\)

\(m\angle P=180 - 64 = 116^{\circ}\)

9.

Step1: Linear pair (supplementary)

\(m\angle2=180 - 113=67^{\circ}\)

10.

Step1: Vertical angles are equal

\(3x + 47=6x-25\)

Step2: Solve for \(x\)

\(47 + 25=6x-3x\), \(3x=72\), \(x = 24\)

Step3: Find \(m\angle SQR\)

\(m\angle SQR=6x-25=6\times24 - 25=144 - 25 = 119^{\circ}\)

11.

Step1: \(\angle DCE+\angle ECB = 90^{\circ}\) (since \(\overline{AB}\perp\overline{CD}\))

\(7x + 2+x + 8=90\)

Step2: Solve for \(x\)

\(8x+10 = 90\), \(8x=80\), \(x = 10\)

Step3: Find \(m\angle DCE\)

\(m\angle DCE=7x + 2=7\times10+2=72^{\circ}\)

12.

Step1: Vertical angles are equal

\(8x-5=4x - 19\) (assuming \(\angle KNM\) and \(\angle MNJ\) are vertical angles, but if they are linear pair \(8x-5+4x - 19=180\))
Assume vertical angles: \(8x-4x=-19 + 5\), \(4x=-14\) (wrong). Assume linear pair: \(12x-24 = 180\), \(12x=204\), \(x = 17\)

Step2: Find \(m\angle KNM\)

\(m\angle KNM=8x-5=8\times17-5=136 - 5=131^{\circ}\)

Answer:

1. \(112^{\circ}\)
2. \(22^{\circ}\)
3. \(56^{\circ}\)
4. \(x = 43^{\circ}\), \(y = 137^{\circ}\), \(z = 137^{\circ}\)
5. \(x = 108^{\circ}\), \(y = 72^{\circ}\), \(z = 18^{\circ}\)
6. \(105^{\circ}\)
7. \(48^{\circ}\)
8. \(116^{\circ}\)
9. \(67^{\circ}\)
10. \(119^{\circ}\)
11. \(72^{\circ}\)
12. \(131^{\circ}\)