Practice Maths

Triangle Properties and Proofs

Key Ideas

Key Terms

Angle sum of a triangle
= 180° (angle sum of triangle).
Exterior angle theorem
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
Isosceles triangle
Two equal sides ⇒ the base angles (opposite the equal sides) are equal.
Equilateral triangle
All sides equal ⇒ all angles = 60°.
Congruent triangles
Have identical shape and size. Tests for congruence:
   – SSS: three pairs of equal sides.
   – SAS: two pairs of equal sides and the included angle equal.
   – AAS: two pairs of equal angles and one pair of equal sides (in corresponding positions).
   – RHS: right angle, hypotenuse equal, one other side equal.
corresponding parts
(sides and angles) of congruent triangles are equal. State: “… (corresponding parts of congruent triangles, ▵ABC ≡ ▵DEF)”.
Hot Tip Always name congruent triangles with vertices in matching order. If ▵ABC ≡ ▵PQR, then A corresponds to P, B to Q, and C to R. This tells you which sides and angles are equal without re-doing the proof.

Worked Example

Question: In the diagram, ▵ABC and ▵DCB share the side BC. AB = DC and AB ∥ DC. Prove ▵ABC ≡ ▵DCB, then find AC if BD = 7 cm.

Step 1 — Set up the proof.
In ▵ABC and ▵DCB:

Step 2 — List equal parts with reasons.
AB = DC   (given)
∠ABC = ∠DCB   (alternate angles, AB ∥ DC)
BC = CB   (common side)

Step 3 — State the congruence test.
∴ ▵ABC ≡ ▵DCB   (SAS)

Step 4 — Use congruence to find the unknown.
AC = BD = 7 cm   (corresponding parts of congruent triangles, ▵ABC ≡ ▵DCB)

The Angle Sum of a Triangle

The three interior angles of any triangle always add up to exactly 180°. This is one of the most used facts in geometry. You can prove it by drawing a line parallel to the base through the apex — the three angles at the apex are alternate angles with the base angles and the middle angle, so they must sum to a straight line (180°).

Using this: if you know two angles of a triangle, find the third by subtracting their sum from 180°.

Exterior Angle Theorem

When you extend one side of a triangle, the angle formed outside the triangle (the exterior angle) equals the sum of the two non-adjacent interior angles (sometimes called the "remote interior angles").

If the interior angles are A, B, and C, and the exterior angle at C is E, then E = A + B.

This follows directly from the angle sum: A + B + C = 180° and E + C = 180° (straight line), so E = A + B.

Isosceles Triangle Properties

An isosceles triangle has two equal sides, and the angles opposite those equal sides are also equal (called the base angles). Key properties:

  • Base angles are equal: if AB = AC, then ∠ABC = ∠ACB.
  • The axis of symmetry (line from apex to midpoint of base) is also the perpendicular bisector of the base and the angle bisector of the apex angle.

In proofs, you can state "base angles of isosceles triangle are equal" as a reason whenever two sides are marked as equal.

Constructing a Logical Triangle Proof

Every proof is a chain of justified steps. For triangle proofs, the most common reasons are:

  • Angle sum of a triangle = 180°
  • Exterior angle equals sum of two non-adjacent interior angles
  • Base angles of an isosceles triangle are equal
  • Vertically opposite angles are equal
  • Angles on a straight line add to 180°

Work step by step: find what you know, apply one rule per step, and cite the reason in brackets.

Strategy: In proof questions, start from what is given and work forward, or start from what you need to prove and work backward, until the two chains meet. Label all angles with letters at the start (e.g., ∠ABC) so your reasoning is unambiguous. Vague statements like "this angle equals that angle" without naming them will not earn full marks.

Mastery Practice

  1. Find the unknown angle(s) in each triangle. Give a reason for each step. Fluency

    1. A triangle has angles 47°, 83°, and x.
    2. An isosceles triangle has one angle of 40° at the apex. Find the two base angles.
    3. An isosceles triangle has base angles of 68° each. Find the apex angle.
    4. An equilateral triangle has one angle expressed as (2x + 10)°. Find x.
    5. A triangle has angles in the ratio 2 : 3 : 4. Find each angle.
    6. Two angles of a triangle are (3x + 5)° and (2x − 10)°. The third angle is 75°. Find x and the sizes of all three angles.

  2. Apply the exterior angle theorem to find each unknown. Fluency

    1. A triangle has interior angles 52° and 61°. Find the exterior angle at the third vertex.
    2. An exterior angle is 130° and one non-adjacent interior angle is 70°. Find the other non-adjacent interior angle.
    3. An exterior angle is (4x + 10)°. The two non-adjacent interior angles are 65° and (2x − 5)°. Find x.
    4. In an isosceles triangle, each base angle is 55°. Find the exterior angle at the apex vertex.

  3. State whether each pair of triangles is congruent. If yes, name the congruence test (SSS, SAS, AAS, or RHS) and write the congruence statement with vertices in matching order. Understanding

    1. ▵ABC with AB = 5, BC = 7, CA = 9 and ▵PQR with PQ = 5, QR = 7, RP = 9.
    2. ▵DEF with DE = 8, ∠DEF = 50°, EF = 6 and ▵GHI with GH = 8, ∠GHI = 50°, HI = 6.
    3. ▵JKL with ∠J = 40°, ∠K = 75°, JK = 10 and ▵MNO with ∠M = 40°, ∠N = 75°, MN = 10.
    4. Two right-angled triangles: hypotenuse 13 cm and one leg 5 cm in each.
    5. ▵PQR with PQ = 6, QR = 8, ∠P = 30° and ▵STU with ST = 6, TU = 8, ∠S = 30°. (Careful — is the angle included?)

  4. Write a full congruence proof for each pair of triangles. Understanding

    1. In quadrilateral ABCD, the diagonals AC and BD bisect each other at M. Prove ▵AMB ≡ ▵CMD.
    2. In ▵PQR, PQ = PR. S is the midpoint of QR. Prove ▵PQS ≡ ▵PRS.
    3. Two triangles share a common base XY. Point Z is above XY and W is below XY such that ZX = ZY and WX = WY. Prove ▵ZXW ≡ ▵ZYW.

  5. Use congruence and triangle properties to find unknowns and justify your answers. Problem Solving

    1. In the figure, AB ∥ DC and AB = DC. The diagonals AC and BD intersect at E. Prove ▵ABE ≡ ▵DCE and hence find BE if DE = 4.5 cm.
    2. In ▵ABC, AB = AC. D is a point on BC such that AD ⊥ BC. Prove that BD = DC, clearly naming each reason. What does this tell you about the perpendicular from the apex of an isosceles triangle?
    3. A surveyor needs to find the distance across a river. She marks points A and B on one side, and a point C directly opposite B on the other side. She walks to D on her side such that D is on the line AC, and measures BD. Explain which congruence test applies and how the river width BC can be determined.
    4. In ▵PQR, the exterior angle at R is 115°. ∠Q = 2∠P. Find ∠P, ∠Q, and ∠R (interior). Verify using the exterior angle theorem.

  6. Multi-step isosceles and angle-chain problems. Problem Solving

    1. In triangle ABC, AB = AC and ∠BAC = 40°. D is a point on BC such that BD = AB. Find ∠ADC, giving a reason for each step.
    2. Triangle PQR has PQ = PR. S is on QR such that PS bisects ∠QPR. The exterior angle at Q is 112°. Find ∠QPS and ∠SPR.
    3. An isosceles triangle has apex angle (4x − 10)° and each base angle (3x + 5)°. Find x and all three angles, then find the exterior angle at the apex vertex.

  7. Congruence in geometric contexts. Problem Solving

    1. In the figure, ABCD is a rectangle. The diagonal BD is drawn. E is a point on BD such that AE = CE. Prove that ▵ABE ≡ ▵CBE and hence find ∠AEB.
    2. Two triangles share a common vertex at P. ▵PAB has PA = 8 cm, AB = 10 cm, ∠PAB = 55°. ▵PCD has PC = 8 cm, CD = 10 cm, ∠PCD = 55°. Prove ▵PAB ≡ ▵PCD, then determine if PB = PD, giving a full reason.
    3. ABCD is a square with diagonals AC and BD intersecting at O. Prove that ▵AOB ≡ ▵BOC, and use this to show that the diagonals of a square bisect the interior angles (i.e., ∠OAB = ∠OBA = 45°).

  8. Exterior angle theorem in multi-step problems. Problem Solving

    1. In ▵ABC, the exterior angle at A is 118°. The exterior angle at B is 135°. Find all three interior angles and hence the exterior angle at C, showing full working.
    2. D is a point on side BC of ▵ABC. ∠ADB is an exterior angle of ▵ADC. If ∠ADB = (3x + 8)°, ∠DAC = (2x − 4)°, and ∠ACD = 52°, find x and all labelled angles. Verify using the exterior angle theorem.
    3. In ▵XYZ, ∠X = 2∠Z and the exterior angle at Y is 105°. Find ∠X, ∠Y (interior), and ∠Z, using the exterior angle theorem and angle sum.

  9. Reasoning with nested and overlapping triangles. Problem Solving

    1. In the diagram, E is a point inside ▵ABC. Lines AE, BE, and CE are drawn. If ∠BAC = 48°, ∠ABC = 62°, and ∠ACB = 70°, and BE extended meets AC at D, find ∠BDC using the exterior angle theorem applied to ▵ABD.
    2. Triangle ABC has D on AB and E on AC such that DE ∥ BC. ∠ADE = 65° and ∠AED = 72°. Find ∠ABC and ∠ACB, giving geometric reasons for each step.
    3. Prove that in any triangle, the sum of the exterior angles (one at each vertex) is always 360°. Present your proof as a series of numbered steps each with a reason.

  10. Congruence proofs and applications in real contexts. Problem Solving

    1. A surveyor marks out two triangular plots of land. Plot 1 has sides 45 m, 62 m, 78 m. Plot 2 has sides 62 m, 45 m, 78 m. Are the plots congruent? Name the test and state whether their areas must be equal. Justify.
    2. In the figure, AB = DE, BC = EF, and ∠ABC = ∠DEF. Prove ▵ABC ≡ ▵DEF, stating clearly the congruence test used. Then find CF if AD = 3 cm and the triangles share vertices B = E and the triangles are positioned so that A, B, E, D are collinear with B and E coincident.
    3. A bridge is reinforced by a triangular truss. Two of the three struts are 5.2 m and 6.8 m long, and the angle between them is 74°. The engineers claim a second identical truss can be built using the same two lengths and the same included angle. State the congruence test that guarantees the second truss will be identical, and explain why the included angle condition is critical (i.e., what can go wrong if the angle is not included).
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