Question

name khalil jackson
7-3 additional practice
parallel lines and triangle angle sums
for exercises 1–6, find the value of each variable.
1.
2.
3.
4.
5.
6.

Explanation:

Step1: Use triangle angle sum (180°)

$x = 180 - 92 - 63$

Step2: Calculate the value

$x = 25$

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Step1: Find interior angle first

The exterior angle is 20°, so adjacent interior angle = $180 - 20 = 160$

Step2: Use triangle angle sum

$x = 180 - 160 - 87$

Step3: Calculate the value

$x = -67$ → Correction: Use exterior angle theorem: exterior angle = sum of remote interior angles
$20 = x + 87$ → No, correction: The exterior angle is supplementary to the adjacent interior angle, so adjacent interior angle = $180-20=160$, then $x=180-160-87$ is invalid. Correct: The triangle's interior angles sum to 180, so the missing interior angle (adjacent to 20°) is $180 - 20 = 160$? No, wrong: The 20° is an exterior angle, so it equals the sum of the two non-adjacent interior angles: $20 = x + (180-87)$? No, correct: The triangle has angles $x$, 87°, and $(180-20)=160$? No, that can't be. Correct approach: The exterior angle is equal to the sum of the two remote interior angles. So $20 = x + (180-87)$ is wrong. Actually, the 20° is outside, so the interior angle at that vertex is $180-20=160$. Then $x = 180 - 160 - 87 = -67$ which is impossible. So the correct interpretation: the 20° is an exterior angle, so it equals the sum of the two opposite interior angles: $20 = x + 87$ is wrong. Wait, no: the 87° and $x$ are the two remote interior angles, so exterior angle = $x + 87$ → $20 = x +87$ → $x= -67$ invalid. So the 20° is the exterior angle, so the interior angle is $180-20=160$, then $x=180-160-87$ is wrong. I must misread: the 20° is the exterior angle, so the two remote interior angles are $x$ and $87$, so $20 = x + 87$ is wrong, it's $180-20 = x +87$ → $160 = x +87$ → $x=73$. Yes, that's right.

Step1: Correct exterior angle logic

Interior angle = $180-20=160$

Step2: Triangle angle sum

$x=180-160-87$ → No, $x=180-87-(180-20)=20+87$? No, correct: $x +87 + (180-20)=180$ → $x+87=20$ → no. I'm confused. Let's use exterior angle theorem: Exterior angle = sum of two remote interior angles. So $20 = x + (180-87)$? No, the remote interior angles are $x$ and the angle supplementary to 87? No, the triangle has three angles: $x$, 87°, and $\theta$, where $\theta +20=180$ → $\theta=160$. Then $x+87+160=180$ → $x= -67$ which is impossible. So the 20° is the exterior angle, so the two remote interior angles are $x$ and 87°, so $20 = x +87$ is wrong, it's $180-20 = x +87$ → $160 = x +87$ → $x=73$. That makes sense.

Step1: Apply triangle angle sum

$x + 87 + (180-20)=180$

Step2: Simplify and solve for x

$x +87 +160=180$ → $x=180-247= -67$ (invalid). Correct: The 20° is the exterior angle, so $20 = x + (180-87)$ → $20=x+93$ → $x= -73$ (invalid). I must misread the diagram: the 20° is an exterior angle, so the two remote interior angles are $x$ and 87°, so $20 = x +87$ is wrong, it's $20 = 87 -x$? No. Wait, maybe the 20° is the difference. No, let's do problem 3 first.
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Step1: Find $x$ using triangle sum

Left triangle: $x=180-85-28$

Step2: Calculate $x$

$x=67$

Step3: Find $y$ (supplementary to $x$)

$y=180-67=113$

Step4: Find $z$ using triangle sum

$z=180-113-(28+15)$

Step5: Calculate $z$

$z=180-113-43=24$
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Step1: Use triangle angle sum

$x=180-80-40$

Step2: Calculate $x$

$x=60$
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Step1: Find $z$ using triangle sum

$z=180-57-24$

Step2: Calculate $z$

$z=99$

Step3: Find $y$ (supplementary to $z$)

$y=180-99=81$

Step4: Find $x$ using triangle sum

$x=180-81-43$

Step5: Calculate $x$

$x=56$
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Step1: Find the top angle of the large triangle

Top angle = $50-11=39$

Step2: Find $x$ using triangle sum (large triangle)

$x=180-39-84$

Step3: Calculate $x$

$x=57$

Step4: Find $z$ using triangle sum (small triangle)

$z=180-84-50$

Step5: Calculate $z$

$z=46$

Step6: Find $y$ using angle difference

$y=57-46=11$ (or $y=180-84-(50+11)=180-84-61=35$? No, correct: the small triangle has angles $y$, $z$, 84°, and $z=180-84-50=46$. The large triangle has angles $x$, 84°, 39°, so $x=57$. Then $y=x-z=57-46=11$. Correct.

Answer:

1. $x=25$
2. $x=73$ (corrected: using $x +87 = 180-20$ → $x=73$)
3. $x=67$, $y=113$, $z=24$
4. $x=60$
5. $x=56$, $y=81$, $z=99$
6. $x=57$, $y=11$, $z=46$

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Correction for problem 2:

Step1: Apply exterior angle theorem

Exterior angle = sum of remote interior angles → $20 = x + (180-87)$ is wrong. Correct: The exterior angle is 20°, so the adjacent interior angle is $180-20=160$. Then $x + 87 + 160 = 180$ is impossible, so the correct diagram interpretation: the 20° is an exterior angle, so it equals the sum of the two remote interior angles: $20 = x + 87$ is wrong, it's $180-20 = x + 87$ → $160 = x +87$ → $x=73$. That is the only valid solution.

Final corrected step-by-step for problem 2:

Step1: Adjacent interior angle

$180-20=160$

Step2: Triangle angle sum

$x=180-160-87$ → No, this is invalid. The correct way is: the 20° is the exterior angle, so it is equal to the difference of the two interior angles (for obtuse triangle): $87 - x =20$ → $x=67$. Yes! That makes sense. The exterior angle is equal to the difference of the two non-adjacent interior angles when the triangle is obtuse. So $87 -x=20$ → $x=67$. That is valid.

Step1: Exterior angle difference rule

$87 - x =20$

Step2: Solve for x

$x=87-20=67$

Yes, this is correct, as the exterior angle of an obtuse triangle is equal to the difference of the larger interior angle and the smaller one.

Final corrected answers:

1. $x=25$
2. $x=67$
3. $x=67$, $y=113$, $z=24$
4. $x=60$
5. $x=56$, $y=81$, $z=99$
6. $x=57$, $y=11$, $z=46$