Question

vertical angles - two angles formed by intersecting lines. vertical angles are equal.
find the missing values in the following.
complementary / supplementary angles/ vertical angles #3 g...... 2007

Explanation:

Problem ①

Step1: Find w (vertical to 80°)

Vertical angles are equal, so $w = 80^\circ$

Step2: Find x (supplementary to w)

Supplementary angles sum to $180^\circ$, so $x = 180^\circ - 80^\circ = 100^\circ$

Step3: Find y (vertical to x)

Vertical angles are equal, so $y = x = 100^\circ$

Step4: Find z (vertical to 80°)

Vertical angles are equal, so $z = 80^\circ$

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Problem ②

Step1: Find w (vertical to 43°)

Vertical angles are equal, so $w = 43^\circ$

Step2: Find x (supplementary to w)

Supplementary angles sum to $180^\circ$, so $x = 180^\circ - 43^\circ = 137^\circ$

Step3: Find y (vertical to x)

Vertical angles are equal, so $y = x = 137^\circ$

Step4: Find z (supplementary to x)

Supplementary angles sum to $180^\circ$, so $z = 180^\circ - 137^\circ = 43^\circ$

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Problem ③

Step1: Find w (vertical to 35°)

Vertical angles are equal, so $w = 35^\circ$

Step2: Find x (complementary to 35°)

Complementary angles sum to $90^\circ$, so $x = 90^\circ - 35^\circ = 55^\circ$

Step3: Find y (vertical to x)

Vertical angles are equal, so $y = x = 55^\circ$

Step4: Find z (right angle)

Perpendicular lines form $90^\circ$, so $z = 90^\circ$

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Problem ④

Step1: Find w (vertical to 40°)

Vertical angles are equal, so $w = 40^\circ$

Step2: Find x (solve for $2x$)

$2x$ and $40^\circ$ are complementary: $2x + 40^\circ = 90^\circ$
$2x = 50^\circ \implies x = 25^\circ$

Step3: Find y (vertical to $2x$)

Vertical angles are equal, so $y = 2x = 50^\circ$

Step4: Find z (supplementary to w)

Supplementary angles sum to $180^\circ$, so $z = 180^\circ - 40^\circ = 140^\circ$

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Problem ⑤

Step1: Find x (solve $5x + 90^\circ = 180^\circ$)

Supplementary angles sum to $180^\circ$: $5x = 90^\circ \implies x = 18^\circ$

Step2: Find w (vertical to $5x$)

$5x = 90^\circ$, so $w = 90^\circ$

Step3: Find y (vertical to $x$)

Vertical angles are equal, so $y = x = 18^\circ$

Step4: Find z (supplementary to w)

Supplementary angles sum to $180^\circ$, so $z = 180^\circ - 90^\circ = 90^\circ$

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Problem ⑥

Step1: Solve for x (vertical angles)

$x + 20^\circ = x^\circ$ is incorrect; use supplementary angles: $x + (x + 20^\circ) = 180^\circ$
$2x + 20^\circ = 180^\circ \implies 2x = 160^\circ \implies x = 80^\circ$

Step2: Find w (vertical to $x$)

Vertical angles are equal, so $w = x = 80^\circ$

Step3: Find y (vertical to $x+20^\circ$)

$x+20^\circ=100^\circ$, so $y = 100^\circ$

Step4: Find z (vertical to w)

Vertical angles are equal, so $z = w = 80^\circ$

Answer:

① $w=80^\circ$, $x=100^\circ$, $y=100^\circ$, $z=80^\circ$
② $w=43^\circ$, $x=137^\circ$, $y=137^\circ$, $z=43^\circ$
③ $w=35^\circ$, $x=55^\circ$, $y=55^\circ$, $z=90^\circ$
④ $w=40^\circ$, $x=25^\circ$, $y=50^\circ$, $z=140^\circ$
⑤ $w=90^\circ$, $x=18^\circ$, $y=18^\circ$, $z=90^\circ$
⑥ $w=80^\circ$, $x=80^\circ$, $y=100^\circ$, $z=80^\circ$