Question

using algebra

1. if ( mangle pqt = (3x + 47)^circ ) and ( mangle sqr = (6x - 25)^circ ), find the measure of ( angle sqr ). diagram of intersecting lines p, t, s, r at q
2. if ( overline{ab} perp overline{cd} ), ( mangle dce = (7x + 2)^circ ) and ( mangle ecb = (x + 8)^circ ), find the measure of ( angle dce ). diagram of ab horizontal, cd vertical at c, e between d and b
3. if ( mangle knm = (8x - 5)^circ ) and ( mangle mnj = (4x - 19)^circ ), find the measure of ( angle knm ). diagram of intersecting lines l, m, k, j at n

Explanation:

Problem 10

Step1: Identify vertical angles

$\angle PQT$ and $\angle SQR$ are vertical angles, so they are equal. Thus, $3x + 47 = 6x - 25$.

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides: \( 47 = 3x - 25 \).
Add 25 to both sides: \( 72 = 3x \).
Divide by 3: \( x = 24 \).

Step3: Find \( m\angle SQR \)

Substitute \( x = 24 \) into \( 6x - 25 \): \( 6(24) - 25 = 144 - 25 = 119 \).

Step1: Use perpendicular lines

Since \( \overline{AB} \perp \overline{CD} \), \( \angle DCE + \angle ECB = 90^\circ \). So, \( (7x + 2) + (x + 8) = 90 \).

Step2: Solve for \( x \)

Combine like terms: \( 8x + 10 = 90 \).
Subtract 10: \( 8x = 80 \).
Divide by 8: \( x = 10 \).

Step3: Find \( m\angle DCE \)

Substitute \( x = 10 \) into \( 7x + 2 \): \( 7(10) + 2 = 70 + 2 = 72 \).

Step1: Identify supplementary angles (linear pair)

$\angle KNM$ and $\angle MNJ$ are supplementary (form a linear pair), so \( (8x - 5) + (4x - 19) = 180 \).

Step2: Solve for \( x \)

Combine like terms: \( 12x - 24 = 180 \).
Add 24: \( 12x = 204 \).
Divide by 12: \( x = 17 \).

Step3: Find \( m\angle KNM \)

Substitute \( x = 17 \) into \( 8x - 5 \): \( 8(17) - 5 = 136 - 5 = 131 \).

Answer:

\( 119^\circ \)

Problem 11