Lines, Angles & Triangles – SAT Math

1. Angle Relationships

Complementary

Sum = 90°

Two angles that add to 90°

Supplementary

Sum = 180°

Two angles that add to 180°

Vertical Angles

Equal

Opposite angles formed by intersecting lines

Linear Pair

Sum = 180°

Adjacent angles on a straight line

📌 RULE: Parallel Lines Cut by a Transversal

When two parallel lines are cut by a transversal:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Alternate exterior angles are equal.
• Same-side interior angles are supplementary (sum to 180°).

Key fact: If two parallel lines are cut by a transversal, all acute angles are equal, and all obtuse angles are equal.

💡 Strategy — Angles

1. Identify the angle relationship (complementary, supplementary, vertical, corresponding).
2. Use the relationship to set up an equation.
3. Solve for the unknown angle.

📝 SOLVED EXAMPLE 1 — Complementary Angles

Two angles are complementary. One angle is 35°. What is the other angle?

Step 1: Complementary angles sum to 90°

Step 2: \(35^\circ + x = 90^\circ\)

Step 3: \(x = 90^\circ - 35^\circ = \color{var(--math)}{55^\circ}\)

✅ \(55^\circ\)

💡 Tip: Complementary → sum = 90°.

📝 SOLVED EXAMPLE 2 — Supplementary Angles

Two angles are supplementary. One angle is 110°. What is the other angle?

Step 1: Supplementary angles sum to 180°

Step 2: \(110^\circ + x = 180^\circ\)

Step 3: \(x = 180^\circ - 110^\circ = \color{var(--math)}{70^\circ}\)

✅ \(70^\circ\)

💡 Tip: Supplementary → sum = 180°.

2. Triangle Properties

📌 RULE: Triangle Angle Sum

The sum of the interior angles of a triangle is 180°.

\[ a + b + c = 180^\circ \]

📌 RULE: Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

For sides \(a\), \(b\), and \(c\):
\[ a + b > c, \quad a + c > b, \quad b + c > a \]

📌 RULE: Exterior Angle Theorem

An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

\[ \text{Exterior Angle} = \text{Angle 1} + \text{Angle 2} \]

Equilateral

All sides equal

All angles = 60°

Isosceles

Two sides equal

Base angles are equal

Scalene

No sides equal

All angles different

Right Triangle

One angle = 90°

Pythagorean theorem applies

📝 SOLVED EXAMPLE 3 — Triangle Angles

A triangle has angles \(50^\circ\) and \(70^\circ\). What is the third angle?

Step 1: Sum of angles = 180°

Step 2: \(50^\circ + 70^\circ + x = 180^\circ\)

Step 3: \(120^\circ + x = 180^\circ\)

Step 4: \(x = \color{var(--math)}{60^\circ}\)

✅ \(60^\circ\)

💡 Tip: Triangle angles always sum to 180°.

3. Triangle Similarity

📌 RULE: Similar Triangles

Two triangles are similar if they have the same shape (corresponding angles are equal).

Criteria:
• AA Similarity: Two corresponding angles are equal.
• SAS Similarity: Two sides are proportional and the included angle is equal.
• SSS Similarity: All three sides are proportional.

Key property: Corresponding sides of similar triangles are proportional.

💡 Strategy — Similar Triangles

1. Identify the corresponding angles.
2. Set up a proportion: \(\frac{\text{side}_1}{\text{side}_2} = \frac{\text{side}_3}{\text{side}_4}\).
3. Cross-multiply and solve for the unknown.

📝 SOLVED EXAMPLE 4 — Similar Triangles

Triangle ABC is similar to triangle DEF. AB = 6, DE = 9, and BC = 8. Find EF.

Step 1: AB/DE = BC/EF (corresponding sides)

Step 2: \(\frac{6}{9} = \frac{8}{x}\)

Step 3: \(6x = 72\)

Step 4: \(x = \color{var(--math)}{12}\)

✅ EF = 12

💡 Tip: Corresponding sides are proportional.

4. Triangle Congruence

📌 RULE: Congruent Triangles

Two triangles are congruent if they have the same size and shape (all corresponding sides and angles are equal).

Criteria:
• SSS: All three sides are equal.
• SAS: Two sides and the included angle are equal.
• ASA: Two angles and the included side are equal.
• AAS: Two angles and a non-included side are equal.
• HL: Hypotenuse and leg of a right triangle are equal.

📝 SOLVED EXAMPLE 5 — Congruence

Triangle ABC has AB = 5, BC = 7, AC = 9. Triangle DEF has DE = 5, EF = 7, DF = 9. Are the triangles congruent?

Step 1: All three sides of ABC match DEF.

Step 2: This is SSS congruence.

Step 3: Yes, the triangles are congruent.

✅ Yes, SSS congruence

💡 Tip: SSS, SAS, ASA, AAS, HL are the congruence criteria.

🧪 Practice Questions

Solve each problem using the rules above. Click "Show Answer" to see the full solution.

Question 1

Two angles are complementary. One angle is \(25^\circ\). What is the other angle?

A) \(25^\circ\)

B) \(55^\circ\)

C) \(75^\circ\)

D) \(65^\circ\)

✓ Answer: D

Complementary: sum = 90°. \(90 - 25 = 65\)

📝 Solution: \(90 - 25 = 65\)

Question 2

Two angles are supplementary. One angle is \(135^\circ\). What is the other angle?

A) \(25^\circ\)

B) \(35^\circ\)

C) \(45^\circ\)

D) \(55^\circ\)

✓ Answer: C

Supplementary: sum = 180°. \(180 - 135 = 45\)

📝 Solution: \(180 - 135 = 45\)

Question 3

A triangle has angles \(45^\circ\) and \(65^\circ\). What is the third angle?

A) \(70^\circ\)

B) \(75^\circ\)

C) \(80^\circ\)

D) \(85^\circ\)

✓ Answer: A

Triangle angles sum to 180°. \(45 + 65 + x = 180\) → \(x = 70\)

📝 Solution: \(180 - 45 - 65 = 70\)

Question 4

Two sides of a triangle are 3 and 7. Which could be the third side?

A) 3

B) 4

C) 5

D) 10

✓ Answer: C

Triangle Inequality: 3 + 4 = 7 (not >), 3 + 5 > 7 ✓, 3 + 7 > 5 ✓, 5 + 7 > 3 ✓

📝 Solution: 3 + 5 = 8 > 7 ✓

Question 5

Triangle ABC is similar to triangle DEF. AB = 4, DE = 6, and AC = 8. Find DF.

A) 10

B) 11

C) 12

D) 12

✓ Answer: D

\(\frac{AB}{DE} = \frac{AC}{DF}\) → \(\frac{4}{6} = \frac{8}{x}\) → \(4x = 48\) → \(x = 12\)

📝 Solution: \(4x = 48\) → \(x = 12\)

Question 6

An exterior angle of a triangle is \(110^\circ\). One of the non-adjacent interior angles is \(45^\circ\). What is the other non-adjacent interior angle?

A) \(55^\circ\)

B) \(65^\circ\)

C) \(75^\circ\)

D) \(85^\circ\)

✓ Answer: B

Exterior angle = sum of non-adjacent interior angles. \(110 = 45 + x\) → \(x = 65\)

📝 Solution: \(110 - 45 = 65\)

Question 7

A triangle has sides 5, 12, and 13. What type of triangle is this?

A) Equilateral

B) Isosceles

C) Scalene

D) Right triangle

✓ Answer: D

\(5^2 + 12^2 = 25 + 144 = 169 = 13^2\) → Pythagorean triple → right triangle

📝 Solution: \(5^2 + 12^2 = 13^2\) → right triangle

Question 8

Two angles are vertical angles. One is \(75^\circ\). What is the other?

A) \(15^\circ\)

B) \(75^\circ\)

C) \(105^\circ\)

D) \(180^\circ\)

✓ Answer: B

Vertical angles are equal. \(75^\circ = 75^\circ\)

📝 Solution: Vertical angles are equal.

Question 9

Triangle ABC has AB = BC. Angle A = \(40^\circ\). What is angle C?

A) \(40^\circ\)

B) \(70^\circ\)

C) \(40^\circ\)

D) \(80^\circ\)

✓ Answer: C

AB = BC → base angles (A and C) are equal. Angle A = 40°, so Angle C = 40°.

📝 Solution: Isosceles triangle → base angles equal.

Question 10

Two parallel lines are cut by a transversal. A corresponding angle is \(50^\circ\). What is the measure of the other corresponding angle?

A) \(40^\circ\)

B) \(50^\circ\)

C) \(130^\circ\)

D) \(180^\circ\)

✓ Answer: B

Corresponding angles are equal when parallel lines are cut by a transversal.

📝 Solution: Corresponding angles are equal.

🎉 WELL DONE!

You've completed the Lines, Angles & Triangles lesson. You now know angle relationships, triangle properties, similarity, and congruence.

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