Triangle Angles Detective Worksheet
Triangle Angles Detective
Cracking the 180° Triangle Angle Sum Rule
Name: Date:
Period: Score:
Detective Secret File: Why Does Every Triangle Equal 180°?
Core Rule
L₁ L₂ 1 2 3 1 3
When a parallel line passes through the top point, the triangle sides act as transversals! The 3 angles line up on a straight flat line: \( \angle 1 + \angle 2 + \angle 3 = 180^\circ \)
Step 1: ADD Add known angles together.
Step 2: SUBTRACT Subtract sum from \(180^\circ\).
Step 3: CHECK Ensure all 3 add to \(180^\circ\).
1 Part 1: Solve for the Missing Interior Angle
Follow the 3-step roadmap to find each missing variable.
Case A: Find \(x\) Acute
65° 45° x
1. Add knowns:
\( 65^\circ + 45^\circ = 110^\circ \)
2. Subtract from \(180^\circ\):
\( x = \)
Case B: Find \(y\) Right Triangle
y 34°
1. Clue: Square = 90°
\( 90^\circ + 34^\circ = 124^\circ \)
2. Subtract from \(180^\circ\):
\( y = \)
Case C: Find \(z\) Isosceles
50° z z
1. Clue: Two equal base angles!
\( 180^\circ - 50^\circ = 130^\circ \) remaining
2. Divide evenly: \( 130^\circ \div 2 \)
\( z = \)
2 Part 2: Busted! Spot the Impossible Triangle
One suspect triangle violates the 180° law. Which one is fake?
Suspect #1 Add them up:
40° + 60° + 80° =
Real (180°) Fake!
Suspect #2 Add them up:
55° + 75° + 60° =
Real (180°) Fake!
Suspect #3 Add them up:
90° + 45° + 45° =
Real (180°) Fake!
Page 1 of 2 • Turn over for the Exterior Angle Master Lab!
Exterior Angle Master Lab
The Remote Interior Secrets & Straight Line Pairs
Exterior Angle Rule: \(\text{Outside} = \text{Remote}_1 + \text{Remote}_2\) OR \(180^\circ - \text{Inside}\)
3 Warm-Up: Spot the Remote Angles (Don't Get Tricked!) "Remote" means far away (opposite), NOT the neighbor next door!
∠A ∠B ∠C ∠x
For exterior angle \(\angle x\):
The 2 remote angles are:
(\(\angle C\) is the neighbor, so it's NOT remote!)
Two Detective Checkpoints:
✓ Method 1: Add the two remote inside angles: \( \angle x = \angle A + \angle B \)
✓ Method 2: Use straight flat line: \( \angle x + \angle C = 180^\circ \)
4 Part 4: Exterior Angle Case Files (Solve for Each Variable)
Notice how exterior angles can point in different directions!
54° 68° d
Case 1: Find \(d\) Remote Sum
Add the 2 opposite inside angles:
\( d = 54^\circ + 68^\circ \)
\( d = \)
48° 66° m
Case 2: Find \(m\) Top Vertex
Top exterior angle equals remote base sum:
\( m = 48^\circ + 66^\circ \)
\( m = \)
75° p 138°
Case 3: Find \(p\) Work Backward
Subtract known remote from exterior:
\( p = 138^\circ - 75^\circ \)
\( p = \)
36° w
Case 4: Find \(w\) Square = 90°
Remember the right angle symbol:
\( w = 90^\circ + 36^\circ \)
\( w = \)
5 Part 5: The Grand Exterior Mystery (Connect Both Methods!)
Master Verification
The Scene: In triangle \(ABC\), base line \(BC\) is extended straight to point \(D\). Follow the clues in order to solve the mystery and verify your work!
A B C D 55° ∠B ∠C_in 125°
Clue 1: Find inside \( \angle B \) Exterior \(125^\circ = 55^\circ + \angle B \implies 125^\circ - 55^\circ\)
\( \angle B = \)
Clue 2: Find inside \( \angle C_{\text{in}} \) Straight line with \(125^\circ \implies 180^\circ - 125^\circ\)
\( \angle C_{\text{in}} = \)
Clue 3: Detective Verification Check Add all 3 inside: \( 55^\circ + \angle B + \angle C_{\text{in}} = \)
(Should = 180°!)
Page 2 of 2 • Brilliant deduction, Detective! You've mastered exterior angles.
