Explanation:
We are given 9 triangle problems where we need to find the value of $ x $. We’ll use two key facts:
1.
Triangle Angle Sum Theorem: The three interior angles of any triangle add up to $ 180^\circ $.
2.
Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.
Let’s solve each one step by step.
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1) Triangle with angles: $ 47^\circ $, $ 58^\circ $, and $ x^\circ $.
Sum = $ 47 + 58 + x = 180 $
→ $ 105 + x = 180 $
→ $ x = 180 - 105 = 75 $
✔ Answer:
75
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2) Triangle with angles: $ x^\circ $, $ 96^\circ $, $ 21^\circ $.
Sum = $ x + 96 + 21 = 180 $
→ $ x + 117 = 180 $
→ $ x = 63 $
✔ Answer:
63
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3) Right triangle (has a $ 90^\circ $ angle), another angle is $ 31^\circ $, and the third angle is labeled $ (3x - 1)^\circ $.
So:
$ 90 + 31 + (3x - 1) = 180 $
→ $ 121 + 3x - 1 = 180 $
→ $ 120 + 3x = 180 $
→ $ 3x = 60 $
→ $ x = 20 $
✔ Answer:
20
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4) This shows an
exterior angle $ x^\circ $, with adjacent interior angles $ 21^\circ $ and $ 34^\circ $.
By Exterior Angle Theorem:
Exterior angle = sum of remote interior angles = $ 21 + 34 = 55 $
✔ Answer:
55
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5) Triangle with angles: $ 100^\circ $ (exterior angle at top), and interior angles $ (2x + 3)^\circ $ and $ 51^\circ $.
Wait — careful! The $ 100^\circ $ is drawn *outside* the triangle, so it's an
exterior angle, and the two interior angles inside the triangle are $ (2x+3)^\circ $ and $ 51^\circ $. So:
Exterior angle = sum of remote interior angles
→ $ 100 = (2x + 3) + 51 $
→ $ 100 = 2x + 54 $
→ $ 2x = 46 $
→ $ x = 23 $
✔ Answer:
23
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6) Triangle with angles: $ 60^\circ $, $ 2x^\circ $, and an exterior angle $ 94^\circ $ at the top vertex.
The exterior angle $ 94^\circ $ is adjacent to the interior angle at that vertex. So the interior angle at that vertex is $ 180 - 94 = 86^\circ $.
Now sum of interior angles:
$ 60 + 2x + 86 = 180 $
→ $ 146 + 2x = 180 $
→ $ 2x = 34 $
→ $ x = 17 $
✔ Answer:
17
(Alternatively, using exterior angle theorem directly: exterior angle = sum of remote interior angles → $ 94 = 60 + 2x $ → $ 2x = 34 $ → $ x = 17 $. Same result.)
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7) Triangle with angles: $ 52^\circ $, $ 43^\circ $, and $ x^\circ $.
Sum: $ 52 + 43 + x = 180 $
→ $ 95 + x = 180 $
→ $ x = 85 $
✔ Answer:
85
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8) Triangle with angles: $ x^\circ $, $ 2x^\circ $, and $ 3x^\circ $.
Sum: $ x + 2x + 3x = 180 $
→ $ 6x = 180 $
→ $ x = 30 $
✔ Answer:
30
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9) Right triangle (has a $ 90^\circ $ angle), other angles: $ (2x - 2)^\circ $ and $ (x + 5)^\circ $.
Sum:
$ 90 + (2x - 2) + (x + 5) = 180 $
→ $ 90 + 2x - 2 + x + 5 = 180 $
→ $ (90 - 2 + 5) + 3x = 180 $
→ $ 93 + 3x = 180 $
→ $ 3x = 87 $
→ $ x = 29 $
✔ Answer:
29
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Now match each answer to the choices given (just to double-check):
1) 75 → in choices
2) 63 → yes
3) 20 → yes
4) 55 → yes
5) 23 → yes
6) 17 → yes
7) 85 → yes
8) 30 → yes
9) 29 → yes
All match.
Final Answer:
1) 75
2) 63
3) 20
4) 55
5) 23
6) 17
7) 85
8) 30
9) 29
Parent Tip: Review the logic above to help your child master the concept of worksheet triangle sum and exterior angle theorem answers.
