Question

1. find the missing measure.
2. find the missing measure.
3. find the missing measures.
4. if the measure of an angle is 13°, find the measure of its supplement.
5. if the measure of an angle is 38°, find the measure of its complement.
6. ∠1 and ∠2 form a linear pair. if m∠1=(5x + 9)° and m∠2=(3x + 11)°, find the measure of each angle.
7. ∠1 and ∠2 are vertical angles. if m∠1=(17x + 1)° and m∠2=(20x - 14)°, find m∠2.
8. ∠k and ∠l are complementary angles. if m∠k=(3x + 3)° and m∠l=(10x - 4)°, find the measure of each angle.
9. if m∠p is three less than twice the measure of ∠q, and ∠p and ∠q are supplementary angles, find each angle measure.
10. if m∠b is two more than three times the measure of ∠c, and ∠b and ∠c are complementary angles, find each angle measure.

Explanation:

Step1: Recall angle - sum properties

For question 1, assume the angles are adjacent and form a right - angle (since one angle is marked with a right - angle symbol). The sum of adjacent angles forming a right - angle is 90°. So, \(x+65 = 90\).

Step2: Solve for \(x\) in question 1

Subtract 65 from both sides of the equation \(x+65 = 90\). We get \(x=90 - 65=25\).

Step3: Recall vertical - angle property for question 2

Vertical angles are equal. So if one angle is 51°, then \(x = 51\).

Step4: Recall angle - sum properties for question 3

The angle \(x\) and 107° are vertical angles, so \(x = 107\). The angles \(y\) and 107° are supplementary (a linear pair), so \(y+107 = 180\), then \(y=180 - 107 = 73\). The angle \(z\) and \(y\) are vertical angles, so \(z = 73\).

Step5: Recall supplementary - angle property for question 4

Two angles are supplementary if their sum is 180°. Let the angle be \(A = 13^{\circ}\), and its supplement \(S\). Then \(A+S = 180\), so \(S=180 - 13=167^{\circ}\).

Step6: Recall complementary - angle property for question 5

Two angles are complementary if their sum is 90°. Let the angle be \(A = 38^{\circ}\), and its complement \(C\). Then \(A + C=90\), so \(C=90 - 38 = 52^{\circ}\).

Step7: Recall linear - pair property for question 6

Since \(\angle1\) and \(\angle2\) form a linear pair, \(m\angle1+m\angle2 = 180\). Substitute \(m\angle1=(5x + 9)\) and \(m\angle2=(3x + 11)\) into the equation: \((5x + 9)+(3x + 11)=180\). Combine like - terms: \(8x+20 = 180\). Subtract 20 from both sides: \(8x=160\). Divide by 8: \(x = 20\). Then \(m\angle1=5x + 9=5\times20+9 = 109^{\circ}\) and \(m\angle2=3x + 11=3\times20+11 = 71^{\circ}\).

Step8: Recall vertical - angle property for question 7

Since \(\angle1\) and \(\angle2\) are vertical angles, \(m\angle1=m\angle2\). So, \(17x + 1=20x-14\). Subtract \(17x\) from both sides: \(1 = 3x-14\). Add 14 to both sides: \(15 = 3x\). Divide by 3: \(x = 5\). Then \(m\angle2=20x-14=20\times5-14 = 86^{\circ}\).

Step9: Recall complementary - angle property for question 8

Since \(\angle K\) and \(\angle L\) are complementary, \(m\angle K+m\angle L = 90\). Substitute \(m\angle K=(3x + 3)\) and \(m\angle L=(10x - 4)\) into the equation: \((3x + 3)+(10x - 4)=90\). Combine like - terms: \(13x-1 = 90\). Add 1 to both sides: \(13x=91\). Divide by 13: \(x = 7\). Then \(m\angle K=3x + 3=3\times7+3 = 24^{\circ}\) and \(m\angle L=10x - 4=10\times7-4 = 66^{\circ}\).

Step10: Recall supplementary - angle property for question 9

Let \(m\angle Q=x\). Then \(m\angle P = 2x-3\). Since \(\angle P\) and \(\angle Q\) are supplementary, \(m\angle P+m\angle Q=180\). Substitute the expressions: \((2x - 3)+x=180\). Combine like - terms: \(3x-3 = 180\). Add 3 to both sides: \(3x=183\). Divide by 3: \(x = 61\). So \(m\angle Q = 61^{\circ}\) and \(m\angle P=2\times61-3=119^{\circ}\).

Step11: Recall complementary - angle property for question 10

Let \(m\angle C=x\). Then \(m\angle B = 3x + 2\). Since \(\angle B\) and \(\angle C\) are complementary, \(m\angle B+m\angle C=90\). Substitute the expressions: \((3x + 2)+x=90\). Combine like - terms: \(4x+2 = 90\). Subtract 2 from both sides: \(4x=88\). Divide by 4: \(x = 22\). So \(m\angle C = 22^{\circ}\) and \(m\angle B=3\times22 + 2=68^{\circ}\).

Answer:

1. \(x = 25\)
2. \(x = 51\)
3. \(x = 107\), \(y = 73\), \(z = 73\)
4. \(167^{\circ}\)
5. \(52^{\circ}\)
6. \(m\angle1 = 109^{\circ}\), \(m\angle2 = 71^{\circ}\)
7. \(m\angle2 = 86^{\circ}\)
8. \(m\angle K = 24^{\circ}\), \(m\angle L = 66^{\circ}\)
9. \(m\angle P = 119^{\circ}\), \(m\angle Q = 61^{\circ}\)
10. \(m\angle B = 68^{\circ}\), \(m\angle C = 22^{\circ}\)