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Angle Investigations

An 8th-grade geometry lesson reviewing foundational angle relationships, triangle angle theorems, transversals, and AA similarity.

LLogan-Greene

Angle Investigations

An 8th-grade geometry lesson reviewing foundational angle relationships, triangle angle theorems, transversals, and AA similarity.

LLogan-Greene

Angle Review Worksheet

Geometry Foundations • 8th Grade Math

Angle Review Worksheet

Part 1: Complementary, Supplementary & Vertical Angles

Name:

Date: Period:

1 Complementary

Angles \(A\) and \(B\) form a right angle. Find the measure of missing angle \(x\).

34° x

Show Work / Equation:

Value of \(x\):

°

2 Supplementary

The two angles lie on a straight line. Find the measure of unknown angle \(y\).

128° y

Show Work / Equation:

Value of \(y\):

°

3 Vertical Angles

Two straight lines intersect at vertex \(P\). Determine the angle measures for \(a\) and \(b\).

52° a b

Angle Reasoning:

\(a\) = ° \(b\) = °

4 Algebra Connection

The two vertical angles shown below are congruent (equal). Solve for \(x\).

x + 25° 75°

Set Up Equation & Solve:

Value of \(x\):

x =

Tip: Always check if your angle looks acute (<90°) or obtuse (>90°) to verify your answer! Page 1 of 3

Geometry Foundations • 8th Grade Math

Angle Review Worksheet

Part 2: Triangle Angle Sum & Exterior Angle Theorem

Name:

5 Triangle Sum

Two interior angles of \(\triangle ABC\) are given. Find the measure of missing angle \(k\).

72° 46° k

Show Work / Equation:

Value of \(k\):

°

6 Right Triangle

Find the measure of missing angle \(m\).

37° m

Show Work / Equation:

Value of \(m\):

°

7 Exterior Angle

Use the Exterior Angle Theorem to find outside angle \(w\).

58° 67° w

Show Work / Equation:

Value of \(w\):

°

8 Find Remote Interior

The exterior angle is 135°. Find the missing inside remote angle \(p\).

65° p 135°

Show Work / Equation:

Value of \(p\):

°

Remember: A straight angle is 180°, so the exterior angle and its adjacent interior angle also add to 180°! Page 2 of 3

Geometry Foundations • 8th Grade Math

Angle Review Worksheet

Part 3: Parallel Transversals & Angle-Angle (AA) Similarity

Name:

9 Parallel Lines Cut by a Transversal

Line \(p \parallel q\)

Lines \(p\) and \(q\) are parallel. Given that Angle 1 = 112°, find each angle and choose the relationship word from the bank.

Word Bank: Vertical • Supplementary • Corresponding • Alternate Interior

p q t 112° ∠2 ∠3 ∠4

Angle Measure Relationship to the 112° Angle

\(\angle 2\)

°

\(\angle 3\)

°

\(\angle 4\)

°

Angle-Angle (AA) Similarity Investigations Find all missing angles first to compare

10 Pair A

Check AA

48° 82° ? Triangle 1 48° ? 50° Triangle 2

T1 missing: \(180^\circ - (48^\circ + 82^\circ) =\) °

T2 missing: \(180^\circ - (48^\circ + 50^\circ) =\) °

Are they similar by AA?

YES NO

Reason:

11 Pair B

Check AA

65° 75° ? Triangle 3 65° ? 45° Triangle 4

T3 missing: \(180^\circ - (65^\circ + 75^\circ) =\) °

T4 missing: \(180^\circ - (65^\circ + 45^\circ) =\) °

Are they similar by AA?

YES NO

Reason:

Great job! Review all your answers and double-check your arithmetic additions & subtractions. Page 3 of 3

Angle Review Answer Key

Geometry Foundations • Teacher Resource

Angle Review Answer Key

Part 1: Complementary, Supplementary & Vertical Angles

Answer Key

1 Complementary (90°)

Angles \(A\) and \(B\) form a right angle. Find the measure of missing angle \(x\).

34° x=56°

Worked Solution:

\(x + 34^\circ = 90^\circ\)

\(x = 90^\circ - 34^\circ = 56^\circ\)

Value of \(x\):

56°

2 Supplementary (180°)

The two angles lie on a straight line. Find the measure of unknown angle \(y\).

128° y=52°

Worked Solution:

\(y + 128^\circ = 180^\circ\)

\(y = 180^\circ - 128^\circ = 52^\circ\)

Value of \(y\):

52°

3 Vertical Angles

Two straight lines intersect at vertex \(P\). Determine the angle measures for \(a\) and \(b\).

52° a=52° b=128°

Worked Reasoning:

\(a = 52^\circ\) (vertical angles are congruent)

\(b = 180^\circ - 52^\circ = 128^\circ\) (supplementary)

\(a\) = 52° \(b\) = 128°

4 Algebra Equation

The two vertical angles shown below are congruent (equal). Solve for \(x\).

x + 25° 75°

Worked Solution:

\(x + 25 = 75\)

\(x = 75 - 25 \implies x = 50\)

Value of \(x\):

x = 50

Teacher Note: Reinforce that vertical angles always form an "X" and are equal across the vertex. Page 1 of 3

Geometry Foundations • Teacher Resource

Angle Review Answer Key

Part 2: Triangle Angle Sum & Exterior Angle Theorem

Answer Key

5 Triangle Sum (180°)

Two interior angles of \(\triangle ABC\) are given. Find the measure of missing angle \(k\).

72° 46° k=62°

Worked Solution:

\(72^\circ + 46^\circ = 118^\circ\)

\(k = 180^\circ - 118^\circ = 62^\circ\)

Value of \(k\):

62°

6 Right Triangle

Find the measure of missing angle \(m\).

37° m=53°

Worked Solution:

Square symbol \(= 90^\circ\)

\(m = 90^\circ - 37^\circ = 53^\circ\) (or \(180 - 127 = 53^\circ\))

Value of \(m\):

53°

7 Exterior Angle

Use the Exterior Angle Theorem to find outside angle \(w\).

Angle Explorer Slides

8th Grade Geometry Review

Angle Investigations

Essential relationships, triangle theorems, parallel transversals, and AA similarity in action.

Angle Pairs

90°, 180° & Vertical

Triangles

Sum & Exterior Angles

Transversals

Parallel Line Rules

Similarity

Angle-Angle Criteria

Part 1 • Angle Foundations

Complementary vs. Supplementary

Sum Check: 90° vs 180°

Complementary Adds to 90°

Two angles combine to form a right angle (\(90^\circ\)).

34° x

\(x + 34^\circ = 90^\circ \implies x = 56^\circ\)

Supplementary Adds to 180°

Two angles combine to form a straight line (\(180^\circ\)).

128° y

\(y + 128^\circ = 180^\circ \implies y = 52^\circ\)

Memory Trick: C (Corner / 90°) comes before S (Straight / 180°)! Slide 2 of 8

Part 1 • Angle Foundations

Vertical Angles & Intersecting Lines

Opposite = Congruent

Intersecting Straight Lines

52° a b 128°

Vertical Angle \(a\)

\(a = 52^\circ\)

Linear Pair \(b\)

\(b = 128^\circ\)

Vertical Angles are Congruent

Angles across from each other are identical (\(\cong\)). When you find one, you immediately know its opposite!

Adjacent Angles are Supplementary

Any two neighboring angles form a straight line: \(52^\circ + b = 180^\circ \implies b = 128^\circ\).

Algebra Example: \(x + 25^\circ = 75^\circ \implies x = 50^\circ\)

Two intersecting lines always produce 2 identical acute angles and 2 identical obtuse angles. Slide 3 of 8

Part 2 • Triangle Theorems

Triangle Angle Sum Theorem

All 3 Angles Sum to 180°

\(\angle A + \angle B + \angle C = 180^\circ\)

General Triangle Problem 5

72° 46° k

Step 1: \(72^\circ + 46^\circ = 118^\circ\)

Step 2: \(180^\circ - 118^\circ = \mathbf{62^\circ}\)

\(k = 62^\circ\)

Right Triangle Shortcut Problem 6

37° m

Shortcut: Acute angles sum to \(90^\circ\)

Calculation: \(90^\circ - 37^\circ = \mathbf{53^\circ}\)

58° 67° w=125°

Worked Solution:

Exterior \(= \text{Remote Interior 1} + \text{Remote Interior 2}\)

\(w = 58^\circ + 67^\circ = 125^\circ\)

Value of \(w\):

125°

8 Remote Interior

The exterior angle is 135°. Find the missing inside remote angle \(p\).

65° p=70° 135°

Worked Solution:

\(p + 65^\circ = 135^\circ\)

\(p = 135^\circ - 65^\circ = 70^\circ\)

Value of \(p\):

70°

Teacher Note: Alternate method for #8: find the adjacent interior angle first (\(180 - 135 = 45^\circ\)), then use triangle sum (\(180 - (65 + 45) = 70^\circ\)). Page 2 of 3

Geometry Foundations • Teacher Resource

Angle Review Answer Key

Part 3: Parallel Transversals & Angle-Angle (AA) Similarity

Answer Key

9 Parallel Lines Cut by a Transversal

Line \(p \parallel q\)

Lines \(p\) and \(q\) are parallel. Given that Angle 1 = 112°, find each angle and choose the relationship word from the bank.

p q t 112° 68° 112° 68°

Angle Measure Relationship to the 112° Angle

\(\angle 2\)

68°

Supplementary

\(\angle 3\)

112°

Alternate Interior

\(\angle 4\)

68°

Supplementary

Angle-Angle (AA) Similarity Investigations — Solutions Teacher Verification Key

10 Pair A

SIMILAR

48° 82° 50° Triangle 1 48° 82° 50° Triangle 2

T1 missing: \(180^\circ - (48^\circ + 82^\circ) =\) 50°

T2 missing: \(180^\circ - (48^\circ + 50^\circ) =\) 82°

Are they similar by AA?

YES NO

Reason: Both triangles share the exact same 3 angle measures (48°, 50°, 82°).

11 Pair B

NOT SIMILAR

65° 75° 40° Triangle 3 65° 70° 45° Triangle 4

T3 missing: \(180^\circ - (65^\circ + 75^\circ) =\) 40°

T4 missing: \(180^\circ - (65^\circ + 45^\circ) =\) 70°

Are they similar by AA?

YES NO

Reason: Only one angle (65°) matches; other angles are 40°/75° vs 45°/70°.

Teacher Key Complete: Both AA similarity investigations (Pair A = Similar, Pair B = Not Similar) fully solved. Page 3 of 3

\(m = 53^\circ\)

Every triangle in flat geometry contains exactly 180 degrees inside. Slide 4 of 8

Part 2 • Triangle Theorems

Exterior Angle Theorem

Exterior = Sum of 2 Remote Angles

58° 67° w 55°

\(w = 58^\circ + 67^\circ = 125^\circ\)

The Shortcut Rule

The outside angle equals the exact sum of the two remote interior angles:

\(\text{Exterior Angle} = \angle A + \angle B\)

Alternative: Linear Pair

Inside angle: \(180^\circ - (58^\circ + 67^\circ) = 55^\circ\).
Straight line: \(180^\circ - 55^\circ = \mathbf{125^\circ}\)!

Working backward? \(\text{Missing Remote} = \text{Exterior} - \text{Known Remote}\). Slide 5 of 8

Part 3 • Transversals

Parallel Lines Cut by a Transversal

Only 2 Angle Measures Exist!

p q t 112° ∠2 ∠3 ∠4

Given \(\angle 1 = 112^\circ\) (Obtuse)

Vertical Angle (\(\angle 2\))

Opposite vertex at same intersection

112°

Alternate Interior (\(\angle 3\))

Opposite sides of transversal inside lines

112°

Supplementary / Linear (\(\angle 4\))

Straight line partner: \(180^\circ - 112^\circ\)

68°

The Big Takeaway: All acute angles are 68°, and all obtuse angles are 112°.

Whenever lines are parallel: All acute angles are equal, all obtuse angles are equal. Slide 6 of 8

Part 3 • Similarity

Angle-Angle (AA) Similarity

2 Matching Angles = Similar (\(\sim\))

Pair A (Problem 10) SIMILAR ✓

Triangle 1

48°, 82°

Missing = 50°

Triangle 2

48°, 50°

Missing = 82°

Both triangles share the same angles:

\{48°, 50°, 82°\}

All angles match \(\implies\) Similar by AA!

Pair B (Problem 11) NOT SIMILAR ✗

Triangle 3

65°, 75°

Missing = 40°

Triangle 4

65°, 45°

Missing = 70°

Comparing all angles:

\{65°, 75°, 40°\} vs \{65°, 45°, 70°\}

Only ONE angle matches \(\implies\) Not Similar!

Pro Tip: NEVER decide similarity until you solve for the 3rd missing angle in both! Slide 7 of 8

Ready to Practice

Your Geometry Toolkit

Worksheet Time!

1. Angle Pairs

• Corner = 90°
• Line = 180°
• Across = Equal

2. Triangles

• 3 angles = 180°
• Right \(\Delta\): acute = 90°
• Exterior = sum of 2 far

3. Transversal

• Acute = Acute
• Obtuse = Obtuse
• Acute + Obtuse = 180°

4. AA Criteria

• Find 3rd angle first
• Check for 2 pairs
• Yes = Similar \(\sim\)

Let's Complete the Worksheet!

Show all equation setups and double-check your arithmetic additions & subtractions.

Slide 8 of 8

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