Question

examples 7 - 9: find the missing angles.
7.
$112^\circ$
$x^\circ$
8.
$124^\circ$
$x^\circ$
9.
$x^\circ$
$y^\circ$
$z^\circ$
$43^\circ$
practice 10 - 12: find the missing angles.
10.
$105^\circ$
$x$
11.
$51^\circ$
$x^\circ$
12.
$x^\circ$
$107^\circ$
$z^\circ$
$y^\circ$
examples 13 - 14: find the value of each variable.
13.
$(6x + 7)^\circ$
$(8x - 17)^\circ$
14.
$(11x - 15)^\circ$
$(5x - 13)^\circ$
practice 15 - 16: find the value of each variable.
15.
$(10x + 1)^\circ$
$(12x - 5)^\circ$
16.
$(4x + 7)^\circ$
$(2x + 5)^\circ$
complementary angles - two angles that add up to $90^\circ$
supplementary angles - two angles that add up to $180^\circ$
adjacent
nonadjacent
adjacent
nonadjacent

Explanation:

Examples 7-9: Find the missing angles

Step1: Use vertical angles theorem

Vertical angles are equal. For problem 7: $x = 112$

Step2: Use supplementary angles rule

Linear pair sums to $180^\circ$. For problem 8: $x = 180 - 124 = 56$

Step3: Use vertical/supplementary rules

For problem9: $x=43$ (vertical angle), $y=180-43=137$, $z=137$ (vertical to $y$)

Step1: Use vertical angles theorem

For problem10: $x = 105$

Step2: Use vertical angles theorem

For problem11: $x = 51$

Step3: Use vertical/supplementary rules

For problem12: $x=180-107=73$, $y=107$ (vertical to $107^\circ$), $z=73$ (vertical to $x$)

Step1: Set vertical angles equal (13)

Vertical angles are congruent: $6x + 7 = 8x - 17$
Solve: $7+17=8x-6x$ → $24=2x$ → $x=12$

Step2: Set supplementary angles equal (14)

Linear pair sums to $180^\circ$: $11x-15 + 5x-13 = 180$
Simplify: $16x -28=180$ → $16x=208$ → $x=13$

Answer:

1. $112^\circ$
2. $56^\circ$
3. $x=43^\circ$, $y=137^\circ$, $z=137^\circ$

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Practice 10-12: Find the missing angles