Question

for each diagram below, identify an angle pair relationship, and use the angle pair relationship to write an algebraic equation. then answer the question. the diagrams are not drawn to scale.

1. what is $m\angle enm$
2. what is $m\angle fgh$
3. what is $m\angle dbc$
4. what is $m\angle npm$

the angle expressions in the diagrams:

• diagram 13: (no clear expression shown)
• diagram 14: $\angle kgf=4x - 5^\circ$, $\angle kgh=3x + 2^\circ$
• diagram 15: $\angle abd=2x - 4^\circ$, $\angle dbc=x + 7^\circ$
• diagram 16: $\angle lpq=x + 28^\circ$, $\angle mpn=x$

Explanation:

Question 14

Step1: Set angles equal (congruent angles)

$4x - 5 = 3x + 2$

Step2: Solve for x

$4x - 3x = 2 + 5$
$x = 7$

Step3: Substitute x to find angle

$m\angle FGH = 3(7) + 2$
$m\angle FGH = 21 + 2 = 23$

Step1: Set angles equal (vertical angles)

$x = x + 28$
Wait, correction: vertical angles are equal, so the angle $x$ and $x+28$ are not vertical. Actually, $\angle NPM$ is vertical to the angle marked $x+28$, and $\angle NPM$ is supplementary to $x$? No, correction: $\angle NPM$ is vertical to the angle labeled $x+28$, so $m\angle NPM = x+28$. First, solve for $x$ using supplementary angles: $x + (x+28) = 180$

Step1: Set supplementary angles sum to 180

$x + (x + 28) = 180$

Step2: Simplify and solve for x

$2x + 28 = 180$
$2x = 180 - 28 = 152$
$x = \frac{152}{2} = 76$

Step3: Find $m\angle NPM$ (vertical to $x+28$)

$m\angle NPM = x + 28 = 76 + 28 = 104$

Step1: Set angles sum to 90 (right angle)

$(2x - 4) + (x + 7) = 90$

Step2: Simplify and solve for x

$3x + 3 = 90$
$3x = 90 - 3 = 87$
$x = \frac{87}{3} = 29$

Step3: Calculate $m\angle DBC$

$m\angle DBC = x + 7 = 29 + 7 = 36$

Answer:

$23^\circ$

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Question 16