(1)
(a) ∠1 ≅ ∠BFD — Vertical angles are congruent.
(b) m∠1 + m∠2 = 90° — Complementary angles sum to 90°.
(c) m∠2 + m∠3 = 90° — Complementary angles sum to 90°.
(d) m∠1 = m∠3 — Both are complementary to ∠2, so they are equal (Congruent Complements Theorem).
(e) m∠1 + m∠4 = 180° — Linear pair (adjacent angles on a straight line).
(f) m∠2 + m∠3 = 90° — Given as complementary.
(g) m∠BFD + m∠3 = 90° — From part (f) and substitution (∠1 ≅ ∠BFD).
(h) m∠2 = m∠4 — Both are supplementary to ∠1 (Supplements of the same angle are congruent).
(i) m∠2 + m∠4 = 180° — Substitution from (h) and (e).
(j) AE ⊥ BD — Because ∠1 and ∠2 form a right angle (90°), so the lines intersect at 90°.
(2)
Given: m∠1 = 155°, m∠2 = 70°
- ∠1 and ∠3 are vertical angles → m∠3 = 155°
- ∠2 and ∠4 are vertical angles → m∠4 = 70°
- ∠1 and ∠5 are adjacent forming a straight line → m∠5 = 180° - 155° = 25°
- ∠2 and ∠6 are adjacent forming a straight line → m∠6 = 180° - 70° = 110°
- ∠5 and ∠7 are vertical angles → m∠7 = 25°
- ∠6 and ∠8 are vertical angles → m∠8 = 110°
Answers:
m∠3 = 155°, m∠4 = 70°, m∠5 = 25°, m∠6 = 110°, m∠7 = 25°, m∠8 = 110°
(3)
Given:
- (3x + 5)° and (x + 15)° are vertical angles → set equal:
3x + 5 = x + 15
2x = 10
x = 5
→ Then (3x + 5) = 20°, (x + 15) = 20°
- (2x + 10)° and (y + 5)° are vertical angles → set equal:
2x + 10 = y + 5
Substitute x = 5:
2(5) + 10 = y + 5
20 = y + 5
y = 15
Answers: x = 5, y = 15
(4)
Let the angle be x.
Its supplement is 180° - x.
Its complement is 90° - x.
Given:
x - 80° = 3(180° - x) - 5(90° - x)
Solve:
x - 80 = 540 - 3x - 450 + 5x
x - 80 = 90 + 2x
x - 2x = 90 + 80
-x = 170
x = -170 → Not possible for an angle measure.
Recheck equation setup:
“An angle 80° less than three times its supplement” → 3(supplement) - 80
“is 50° more than five times its complement” → 5(complement) + 50
So:
3(180 - x) - 80 = 5(90 - x) + 50
Left: 540 - 3x - 80 = 460 - 3x
Right: 450 - 5x + 50 = 500 - 5x
Equation:
460 - 3x = 500 - 5x
2x = 40
x = 20
Check:
Angle = 20°
Supplement = 160°, 3×160 = 480, minus 80 = 400
Complement = 70°, 5×70 = 350, plus 50 = 400 → Equal.
Answer: 20°
Parent Tip: Review the logic above to help your child master the concept of complementary and supplementary angle worksheet.
