Geometric proof

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1. Isosceles triangles and perpendicular bisectors

Warm-up

Three quick questions from earlier lessons.

Discover

An isosceles triangle has two equal sides. Fold it along the line from the apex to the midpoint of the base. What do you notice about the base angles?

  1. They are equal
  2. One is twice the other

Change the height of the apex. Compare the base angles.

  1. They are always equal
  2. One grows faster than the other

Key idea

Symmetry gives equal parts.

In an isosceles triangle the base angles are equal. A point is on the perpendicular bisector of ABAB exactly when it is equally far from AA and BB.

Worked example

Apex angle 40∘40^\circ.

Base angles together: 140∘140^\circ
180 − 40.
Each base angle: 70∘70^\circ
Equal halves.

Your turn, guided

Your turn: base angles 35∘35^\circ each. The apex angle.

110∘110^\circ
180 − 70.

Practice ★

  1. An isosceles triangle has apex angle 114∘114^\circ (between the two equal sides). Find each base angle, in degrees.
  2. An isosceles triangle has apex angle 38∘38^\circ (between the two equal sides). Find each base angle, in degrees.
  1. Each base angle of an isosceles triangle measures 52∘52^\circ. Find the apex angle, in degrees.

Watch out!

Practice ★★

  1. Point PP lies on the perpendicular bisector of ABAB, and PA=13PA = 13. Find PBPB.Not to scale
  2. Point PP lies on the perpendicular bisector of ABAB, and PA=9PA = 9. Find PBPB.Not to scale

Exit ticket

Three questions to check you have got it.

  1. An isosceles triangle has apex angle 114∘114^\circ (between the two equal sides). Find each base angle, in degrees.
  2. An isosceles triangle has apex angle 38∘38^\circ (between the two equal sides). Find each base angle, in degrees.
  3. An isosceles triangle has apex angle 129∘129^\circ (between the two equal sides). Find each base angle, in degrees.

Summary card

Isosceles triangles and bisectors

Base angles equal; perpendicular bisector = points equidistant from the ends.

Watch out: halve 180° − apex for each base angle.

2. Properties of parallelograms

Warm-up

Three quick questions from earlier lessons.

Discover

A diagonal cuts a parallelogram into two triangles. Are they congruent?

  1. Yes
  2. Only if it is a rectangle

Key idea

Properties of parallelograms.

In a parallelogram: opposite sides are equal, opposite angles are equal, consecutive angles add to 180∘180^\circ, and the diagonals bisect each other.

Worked example

Parallelogram with ∠A=72∘\angle A = 72^\circ and diagonal BD=18BD = 18.

∠C=72∘\angle C = 72^\circ
Opposite angle.
∠B=∠D=108∘\angle B = \angle D = 108^\circ
Consecutive.
BM=9BM = 9
Diagonals bisect each other.

Your turn, guided

Your turn: ∠B=125∘\angle B = 125^\circ. Find ∠C\angle C.

55∘55^\circ
Consecutive: 180 − 125.

Practice ★

  1. In parallelogram ABCDABCD, ∠A=42∘\angle A = 42^\circ. Find ∠B\angle B, in degrees.
  2. In parallelogram ABCDABCD, ∠A=139∘\angle A = 139^\circ. Find ∠B\angle B, in degrees.
  1. The diagonals of parallelogram ABCDABCD meet at MM, and AC=18AC = 18. Find AMAM.Not to scale

Watch out!

Practice ★★

  1. In parallelogram ABCDABCD, AB=2x+3AB = 2x + 3 and CD=x+8CD = x + 8. Find xx.Not to scale
  1. Boss problem. In parallelogram ABCDABCD, ∠A=(3x+9)∘\angle A = (3x + 9)^\circ and ∠C=(x+45)∘\angle C = (x + 45)^\circ.

Exit ticket

Three questions to check you have got it.

  1. In parallelogram ABCDABCD, ∠A=42∘\angle A = 42^\circ. Find ∠B\angle B, in degrees.
  2. In parallelogram ABCDABCD, ∠A=139∘\angle A = 139^\circ. Find ∠B\angle B, in degrees.
  3. In parallelogram ABCDABCD, ∠A=41∘\angle A = 41^\circ. Find ∠B\angle B, in degrees.

Summary card

Parallelograms

Opposite sides and angles equal; consecutive angles 180°; diagonals bisect each other.

Watch out: opposite (equal) vs consecutive (supplementary).

3. Interior and exterior angles of polygons

Warm-up

Three quick questions from earlier lessons.

Discover

A triangle's angles add up to 180°. What do a quadrilateral's angles add up to?

  1. 360°
  2. 180°

Key idea

Interior angles come from triangles; exterior angles always total 360°.

From one vertex, diagonals split an nn-sided polygon into n−2n - 2 triangles, so its interior angles add up to (n−2)⋅180°(n - 2) \cdot 180°. Walking around any convex polygon you turn through 360° in all: the exterior angles add up to 360°. In a regular polygon all of them are equal.

Worked example

A regular hexagon.

(6−2)⋅180°=720°(6 - 2) \cdot 180° = 720°
Interior sum.
720°÷6=120°720° \div 6 = 120°
Each interior angle.
360°÷6=60°360° \div 6 = 60°
Each exterior angle.

Your turn, guided

Your turn: the interior angle sum of a pentagon.

540°540°
(5 − 2) · 180.

Practice ★

  1. What is the sum of the interior angles of a polygon with 12 sides, in degrees?
  2. What is the sum of the interior angles of a polygon with 9 sides, in degrees?
  1. Each exterior angle of a regular polygon with 3 sides measures how many degrees?Not to scale

Watch out!

Practice ★★

  1. The interior angles of a polygon add up to 720°. How many sides does the polygon have?
  2. The interior angles of a polygon add up to 900°. How many sides does the polygon have?
  1. Each exterior angle of a regular polygon with 3 sides measures how many degrees?Not to scale

Exit ticket

Three questions to check you have got it.

  1. What is the sum of the interior angles of a polygon with 12 sides, in degrees?
  2. What is the sum of the interior angles of a polygon with 9 sides, in degrees?
  3. What is the sum of the interior angles of a polygon with 6 sides, in degrees?

Summary card

Polygon angles

Interior sum (n − 2) · 180°; exterior sum 360°.

Watch out: n − 2 triangles, not n.

4. Writing congruence proofs

Warm-up

Three quick questions from earlier lessons.

Discover

To use SAS in a proof, what must you have shown first?

  1. Two pairs of equal sides and the equal angles between them
  2. That the triangles are congruent

Key idea

A proof is a chain of statements, each with a reason.

Typical chain: state the given facts → derive new facts (vertical angles, midpoints, parallel lines, shared sides) → apply a criterion (SSS, SAS, ASA, AAS) → conclude, then use corresponding parts of the congruent triangles.

Worked example

MM is the midpoint of ADAD and of BCBC.

AM=MDAM = MD, BM=MCBM = MC
Definition of midpoint.
∠AMB=∠DMC\angle AMB = \angle DMC (vertical angles)
They are opposite at M.
△AMB≅△DMC\triangle AMB \cong \triangle DMC, so AB=DCAB = DC
SAS.

Your turn, guided

Your turn: how many statements (facts) does SAS need before the conclusion?

33
Side, angle, side.

Practice ★

  1. In a proof, two segments of length 3 cross at MM. Which reason justifies ∠AMB=∠DMC\angle AMB = \angle DMC?Not to scale
    1. Vertical angles are equal
    2. Corresponding angles of parallel lines
    3. Corresponding parts of congruent triangles
  1. Segments ADAD and BCBC cross at their common midpoint MM (so AM=MDAM = MD and BM=MCBM = MC, each pair 6 cm). Put the steps proving △AMB≅△DMC\triangle AMB \cong \triangle DMC in order.Not to scale
    • AM=MDAM = MD and BM=MCBM = MC (M is the midpoint)
    • ∠AMB=∠DMC\angle AMB = \angle DMC (vertical angles)
    • Two sides and the included angle match: SAS
    • So △AMB≅△DMC\triangle AMB \cong \triangle DMC

Watch out!

Practice ★★

  1. Two triangles have two equal angles (54∘54^\circ and 70∘70^\circ) and an equal side that is not between them. Which criterion applies?
    1. AAS (equivalent to ASA, since the third angles are equal too)
    2. SSA
    3. None: they need not be congruent

Exit ticket

Three questions to check you have got it.

  1. In a proof, two segments of length 3 cross at MM. Which reason justifies ∠AMB=∠DMC\angle AMB = \angle DMC?Not to scale
    1. Vertical angles are equal
    2. Corresponding angles of parallel lines
    3. Corresponding parts of congruent triangles
  2. In a proof, two segments of length 6 cross at MM. Which reason justifies ∠AMB=∠DMC\angle AMB = \angle DMC?Not to scale
    1. Vertical angles are equal
    2. Corresponding angles of parallel lines
    3. Corresponding parts of congruent triangles
  3. In a proof, two segments of length 9 cross at MM. Which reason justifies ∠AMB=∠DMC\angle AMB = \angle DMC?Not to scale
    1. Vertical angles are equal
    2. Corresponding angles of parallel lines
    3. Corresponding parts of congruent triangles

Summary card

Congruence proofs

Facts with reasons → criterion → conclusion → corresponding parts.

Watch out: never use what you are proving as a reason.

Answers: Geometric proof

1. Isosceles triangles and perpendicular bisectors

A

  1. 3333
  2. 7171

B

  1. 7676

C

  1. 1313
  2. 99

D

  1. 3333
  2. 7171
  3. 512\frac{51}{2}

2. Properties of parallelograms

A

  1. 138138
  2. 4141

B

  1. 99

C

  1. 55

D

  1. 117117

E

  1. 138138
  2. 4141
  3. 139139

3. Interior and exterior angles of polygons

A

  1. 18001800
  2. 12601260

B

  1. 120120

C

  1. 66
  2. 77

D

  1. 120120

E

  1. 18001800
  2. 12601260
  3. 720720

4. Writing congruence proofs

A

  1. Vertical angles are equal

B

  1. 1. AM=MDAM = MD and BM=MCBM = MC (M is the midpoint) 2. ∠AMB=∠DMC\angle AMB = \angle DMC (vertical angles) 3. Two sides and the included angle match: SAS 4. So △AMB≅△DMC\triangle AMB \cong \triangle DMC

C

  1. AAS (equivalent to ASA, since the third angles are equal too)

D

  1. Vertical angles are equal
  2. Vertical angles are equal
  3. Vertical angles are equal