Question

1. what is the value of x, y and z?
2. what is the value of x?
3. use the graph to the right to find the value of each angle. show your thinking.

∠zay= ____ ∠zax = ____
∠uaw = ____ ∠xaw= ____

1. solve for x and fill in all the missing angles. show your thinking.

Explanation:

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Problem 14: Find x, y, z

Step1: Find x (supplementary to 76°)

Angles on a line sum to 180°.
$x = 180^\circ - 76^\circ$

Step2: Find y (vertical to 76°)

Vertical angles are equal.
$y = 76^\circ$

Step3: Find z (vertical to x)

Vertical angles are equal.
$z = x = 104^\circ$

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Problem 15: Find x

Step1: Sum angles on a line to 180°

Angles on a straight line sum to 180°.
$x + 55^\circ + 40^\circ = 180^\circ$

Step2: Solve for x

Isolate x by subtracting known angles.
$x = 180^\circ - 55^\circ - 40^\circ$

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Problem 16: Find all angles

Step1: Find ∠ZAY

Right angle (marked by square) = 90°.
$\angle ZAY = 90^\circ$

Step2: Find ∠ZAX

Supplementary to 50°, sum to 180°.
$\angle ZAX = 180^\circ - 90^\circ - 50^\circ = 40^\circ$

Step3: Find ∠UAW

Vertical to ∠ZAX + right angle.
$\angle UAW = 90^\circ + 40^\circ = 130^\circ$

Step4: Find ∠XAW

Vertical to 50°, vertical angles equal.
$\angle XAW = 50^\circ$

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Problem 17: Solve for x and missing angles

Left Diagram

Step1: Set up supplementary angle equation

Angles on a line sum to 180°.
$45^\circ + (5x + 25)^\circ = 180^\circ$

Step2: Simplify and solve for x

Combine constants, isolate x.
$5x + 70 = 180 \implies 5x = 110 \implies x = 22$

Step3: Calculate $(5x+25)^\circ$

Substitute x=22 into the expression.
$5(22) + 25 = 135^\circ$

Step4: Find vertical angles

Vertical angles are equal to 45° and 135°.

Right Diagram

Step1: Set up vertical angle equation

Vertical angles are equal.
$(3x + 7)^\circ = 121^\circ$

Step2: Solve for x

Isolate x by subtracting 7, divide by 3.
$3x = 114 \implies x = 38$

Step3: Calculate supplementary angle

Supplementary to 121°, sum to 180°.
$180^\circ - 121^\circ = 59^\circ$

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Answer:

Problem 14

$x = 104^\circ$, $y = 76^\circ$, $z = 104^\circ$

Problem 15

$x = 85^\circ$

Problem 16

$\angle ZAY = 90^\circ$, $\angle ZAX = 40^\circ$, $\angle UAW = 130^\circ$, $\angle XAW = 50^\circ$

Problem 17

Left diagram: $x=22$, missing angles: $135^\circ$, $45^\circ$, $135^\circ$
Right diagram: $x=38$, missing angles: $121^\circ$, $59^\circ$, $59^\circ$