Question

use the exterior angle theorem to find the measure of each angle in degrees.
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Explanation:

Step1: Recall the Exterior - Angle Theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.

Problem 7

For the triangle with exterior angle $\angle ABC = 147^{\circ}$, non - adjacent interior angles are $\angle C=6x^{\circ}$ and $\angle D=(9x - 3)^{\circ}$.

Step2: Set up the equation

$6x+(9x - 3)=147$.
Combining like terms, we get $15x-3 = 147$.
Adding 3 to both sides: $15x=150$.
Dividing both sides by 15, $x = 10$.

Step3: Find the angle measures

$m\angle C=6x=6\times10 = 60^{\circ}$.
$m\angle D=9x - 3=9\times10-3 = 87^{\circ}$.
$m\angle DBC = 147^{\circ}$ (given exterior angle).

Problem 8

For the triangle with exterior angle $\angle TUW = 99^{\circ}$, non - adjacent interior angles are $\angle T = 3x^{\circ}$ and $\angle W=6x^{\circ}$.

Step4: Set up the equation

$3x+6x=99$, so $9x=99$.
Dividing both sides by 9, $x = 11$.

Step5: Find the angle measures

$m\angle T=3x=3\times11 = 33^{\circ}$.
$m\angle W=6x=6\times11 = 66^{\circ}$.
$m\angle TUW = 99^{\circ}$ (given exterior angle).

Problem 9

For the triangle with exterior angle $\angle SRQ=141^{\circ}$, non - adjacent interior angles are $\angle P=(6x - 3)^{\circ}$ and $\angle Q = 10x^{\circ}$.

Step6: Set up the equation

$(6x - 3)+10x=141$.
Combining like terms, $16x-3 = 141$.
Adding 3 to both sides, $16x=144$.
Dividing both sides by 16, $x = 9$.

Step7: Find the angle measures

$m\angle P=6x - 3=6\times9-3 = 51^{\circ}$.
$m\angle Q=10x=10\times9 = 90^{\circ}$.
$m\angle PRQ=180 - 141=39^{\circ}$ (since the exterior angle and its adjacent interior angle are supplementary).

Problem 10

For the triangle with exterior angle $\angle EFG = 126^{\circ}$, non - adjacent interior angles are $\angle D = 4x^{\circ}$ and $\angle E=(7x - 6)^{\circ}$.

Step8: Set up the equation

$4x+(7x - 6)=126$.
Combining like terms, $11x-6 = 126$.
Adding 6 to both sides, $11x=132$.
Dividing both sides by 11, $x = 12$.

Step9: Find the angle measures

$m\angle D=4x=4\times12 = 48^{\circ}$.
$m\angle E=7x - 6=7\times12-6 = 78^{\circ}$.
$m\angle DFE=180 - 126=54^{\circ}$ (since the exterior angle and its adjacent interior angle are supplementary).

Problem 11

For the triangle with exterior angle $\angle KLM = 107^{\circ}$, non - adjacent interior angles are $\angle J=(5x + 9)^{\circ}$ and $\angle K = 9x^{\circ}$.

Step10: Set up the equation

$(5x + 9)+9x=107$.
Combining like terms, $14x+9 = 107$.
Subtracting 9 from both sides, $14x=98$.
Dividing both sides by 14, $x = 7$.

Step11: Find the angle measures

$m\angle J=5x + 9=5\times7+9 = 44^{\circ}$.
$m\angle K=9x=9\times7 = 63^{\circ}$.
$m\angle KLJ=180 - 107=73^{\circ}$ (since the exterior angle and its adjacent interior angle are supplementary).

Problem 12

For the triangle with exterior angle $\angle DEF = 121^{\circ}$, non - adjacent interior angles are $\angle C=(13x + 19)^{\circ}$ and $\angle D=(12x + 2)^{\circ}$.

Step12: Set up the equation

$(13x + 19)+(12x + 2)=121$.
Combining like terms, $25x+21 = 121$.
Subtracting 21 from both sides, $25x=100$.
Dividing both sides by 25, $x = 4$.

Step13: Find the angle measures

$m\angle C=13x + 19=13\times4+19 = 71^{\circ}$.
$m\angle D=12x + 2=12\times4+2 = 50^{\circ}$.
$m\angle DEC=180 - 121=59^{\circ}$ (since the exterior angle and its adjacent interior angle are supplementary).

Answer:

Problem 7

$m\angle C = 60^{\circ}$, $m\angle D = 87^{\circ}$, $m\angle DBC = 147^{\circ}$

Problem 8

$m\angle T = 33^{\circ}$, $m\angle W = 66^{\circ}$, $m\angle TUW = 99^{\circ}$

Problem 9

$m\angle P = 51^{\circ}$, $m\angle Q = 90^{\circ}$, $m\angle PRQ = 39^{\circ}$

Problem 10

$m\angle D = 48^{\circ}$, $m\angle E = 78^{\circ}$, $m\angle DFE = 54^{\circ}$

Problem 11

$m\angle J = 44^{\circ}$, $m\angle K = 63^{\circ}$, $m\angle KLJ = 73^{\circ}$

Problem 12

$m\angle C = 71^{\circ}$, $m\angle D = 50^{\circ}$, $m\angle DEC = 59^{\circ}$