Ext Angle THM Practice Triangle Sum Practice PDF | PDF | Geometric ... - Free Printable

Let’s solve each problem one by one using the Exterior Angle Theorem and the Triangle Sum Theorem.

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🔢 Key Rules to Remember:

- Triangle Sum Theorem: The three interior angles of a triangle add up to 180°.
- Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two opposite (remote) interior angles.

Also, remember:
→ A straight line = 180°
→ If you see an angle next to another on a straight line, they are supplementary (add to 180°).

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## Problem 1:
We have triangle DCE.
Angle at D = 70°, angle at E = 24°.
We need to find angle at C (marked with ?), but note — it's the *exterior* angle at C? Wait — looking at diagram: point G-C-E is a straight line. So angle GCD is the exterior angle at C.

Actually, in triangle DCE, angles inside are:
- ∠D = 70°
- ∠E = 24°
So ∠DCE (interior) = 180 - 70 - 24 = 86°

But the question mark is at ∠GCD — which is adjacent to ∠DCE on straight line GCE → so ∠GCD + ∠DCE = 180°
→ ∠GCD = 180 - 86 = 94°

✔ But wait — maybe they want the exterior angle directly? Let’s check Exterior Angle Theorem:
The exterior angle at C (∠GCD) should equal sum of remote interior angles: ∠D + E = 70 + 24 = 94° → same answer!

✔️ Answer for #1: 94°

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## Problem 2:
Triangle HGF. Angles given: ∠H = 82°, ∠G = 42°. We need ∠F (marked ?). But note — point F-A is extended, so we’re being asked for the exterior angle at F? Or interior?

Looking at diagram: points H-F-A are colinear? Actually, from diagram: side HF is extended to A, so angle at F marked with ? is the exterior angle at F.

Interior angles of triangle: ∠H = 82°, ∠G = 42° → so interior ∠F = 180 - 82 - 42 = 56°

Then exterior angle at F = 180 - 56 = 124°?
OR use Exterior Angle Theorem: exterior angle = sum of two remote interior angles → ∠H + G = 82 + 42 = 124°

✔️ Answer for #2: 124°

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## Problem 3:
Triangle TUS. Exterior angle at U is 123° (on line A-U-S). Interior angle at S is 69°. Find ∠T (?).

Use Exterior Angle Theorem:
Exterior angle at U = ∠T + ∠S
→ 123 = ∠T + 69
→ ∠T = 123 - 69 = 54°

✔️ Answer for #3: 54°

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## Problem 4:
Right triangle TUR (right angle at U). Exterior angle at S (on line E-S-U) is 150°. Need to find ∠T (?).

First, since E-S-U is straight, interior angle at S (in triangle) = 180 - 150 = 30°

Triangle has right angle at U → 90°, angle at S = 30°, so angle at T = 180 - 90 - 30 = 60°

Alternatively, using exterior angle theorem:
Exterior angle at S = ∠T + U → 150 = ∠T + 90 → ∠T = 60°

✔️ Answer for #4: 60°

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## Problem 5:
Triangle DCB. Given: ∠C = 49°, exterior angle at B (along Y-B-C) is 143°. Find ∠D (?).

Exterior angle at B = ∠D + ∠C
→ 143 = ∠D + 49
→ ∠D = 143 - 49 = 94°

✔️ Answer for #5: 94°

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## Problem 6:
Triangle ABC. Given: ∠B = 34°, exterior angle at A (along P-A-C) is 150°. Find ∠ACB (?) — that’s angle at C.

Wait — let’s label carefully. Points: P-A-C is straight line → exterior angle at A is 150°. That means interior angle at A = 180 - 150 = 30°.

In triangle ABC:
∠A = 30°, ∠B = 34° → ∠C = 180 - 30 - 34 = 116°

But wait — the question mark is at angle ACB? Yes — that’s angle at C → 116°

Alternatively, using exterior angle theorem:
Exterior angle at A = ∠B + C → 150 = 34 + ∠C → ∠C = 116°

✔️ Answer for #6: 116°

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Now moving to “Solve for x” problems.

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## Problem 7:
Triangle CDB. Exterior angle at D (along P-D-B) is labeled as 17x - 1.
Interior angles: ∠C = 84°, ∠B = 7x + 5.

By Exterior Angle Theorem:
Exterior angle at D = ∠C + B
→ 17x - 1 = 84 + (7x + 5)

Solve:

17x - 1 = 84 + 7x + 5
17x - 1 = 89 + 7x
Subtract 7x both sides:
10x - 1 = 89
Add 1:
10x = 90
Divide by 10:
x = 9

Check:
Left: 17(9)-1 = 153 - 1 = 152
Right: 84 + 7(9)+5 = 84 + 63 + 5 = 152 → ✔️

✔️ Answer for #7: x = 9

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## Problem 8:
Triangle EFG. Exterior angle at E (along J-E-G) is 130°.
Interior angles: ∠F = 16x + 4, ∠G = 14x + 6.

Exterior angle at E = ∠F + G
→ 130 = (16x + 4) + (14x + 6)

Simplify:

130 = 30x + 10
Subtract 10:
120 = 30x
Divide by 30:
x = 4

Check:
∠F = 16(4)+4 = 64+4=68
∠G = 14(4)+6=56+6=62
Sum = 68+62=130 → matches exterior angle ✔️

✔️ Answer for #8: x = 4

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## Problem 9:
Triangle UTG. Right? No — just triangle.
Angles: ∠U = 84°, ∠T = 10x - 3, exterior angle at G (along S-G-U) is 21x + 4.

Note: exterior angle at G = ∠U + ∠T
→ 21x + 4 = 84 + (10x - 3)

Solve:

21x + 4 = 84 + 10x - 3
21x + 4 = 81 + 10x
Subtract 10x:
11x + 4 = 81
Subtract 4:
11x = 77
Divide by 11:
x = 7

Check:
∠T = 10(7)-3 = 70-3=67
∠U = 84
Sum = 67+84=151
Exterior angle = 21(7)+4=147+4=151 → ✔️

✔️ Answer for #9: x = 7

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## Problem 10:
Triangle TSR. Angles: ∠T = 85°, ∠S = 5x - 10, exterior angle at R (along J-R-S) is 11x + 3.

Exterior angle at R = ∠T + ∠S
→ 11x + 3 = 85 + (5x - 10)

Solve:

11x + 3 = 85 + 5x - 10
11x + 3 = 75 + 5x
Subtract 5x:
6x + 3 = 75
Subtract 3:
6x = 72
Divide by 6:
x = 12

Check:
∠S = 5(12)-10=60-10=50
∠T = 85
Sum = 50+85=135
Exterior angle = 11(12)+3=132+3=135 → ✔️

✔️ Answer for #10: x = 12

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## ✔ Final Answers:

1) 94°
2) 124°
3) 54°
4) 60°
5) 94°
6) 116°
7) x = 9
8) x = 4
9) x = 7
10) x = 12