7. Triangles

7.1 Introduction - Learning Objectives

Mathematical symbols used in Gerometry

Types of Triangles.

7.2 Congruence of Triangles

7.3 Criteria for Congruence of Triangles

Axiom 7.1 (SAS Congruence Rule)

Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle.

Example 1:

In Fig. 7.8, OA = OB and OD = OC. Show that

(i) ∆ AOD ≅ ∆ BOC and (ii) AD || BC.

Example 2:

AB is a line segment and line l is its perpendicular bisector. Prove that any point P on l is equidistant from A and B.

Theorem 7.1 (ASA Congruence Rule)

Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle.

Example 3:

Given AB || CD, O is the midpoint of AD. Show that ∆AOB ≅ ∆DOC and O is the midpoint of BC.

Exercise 7.1

Show that ∆ABC ≅ ∆ABD in quadrilateral ACBD where AC = AD and AB bisects ∠A. What can you say about BC and BD?
Prove ∆ABD ≅ ∆BAC and deduce BD = AC, ∠ABD = ∠BAC in quadrilateral ABCD with AD = BC and ∠DAB = ∠CBA.
Show that CD bisects AB given AD and BC are equal perpendiculars to AB.

Prove ∆ABC ≅ ∆CDA when l, m are intersected by parallel lines p, q.
Show ∆APB ≅ ∆AQB and hence BP = BQ using angle bisector l and perpendiculars.
Show BC = DE using given conditions in triangle ABC.

Prove ∆DAP ≅ ∆EBP and hence AD = BE.
Prove ∆AMC ≅ ∆BMD and ∆DBC ≅ ∆ACB in right triangle ABC where M is midpoint of hypotenuse and DM = CM.

7.4 Some Properties of a Triangle

Theorem 7.2

Angles opposite to equal sides of an isosceles triangle are equal.

Theorem 7.3

Sides opposite to equal angles of a triangle are equal.

Example 4:

Show that AB = AC and ∆ABC is isosceles if the bisector of ∠A is perpendicular to BC.

Example 5:

Show BF = CE given E and F are midpoints of equal sides AB and AC in triangle ABC.

Example 6:

Show that AD = AE given AB = AC and BE = CD in an isosceles triangle ABC.

Exercise 7.2

Prove OB = OC and AO bisects ∠A using angle bisectors in an isosceles triangle.
Show that triangle ABC is isosceles with AB = AC when AD is perpendicular bisector of BC.
Prove BE = CF when drawn from equal sides AC and AB.
Prove triangle is isosceles when BE and CF are equal altitudes.
Prove ∠ABD = ∠ACD in isosceles triangles ABC and DBC on same base BC.
Prove ∠BCD = 90° using given data.
Find ∠B and ∠C in right angled isosceles triangle with ∠A = 90° and AB = AC.
Prove angles of an equilateral triangle are 60° each.

7.5 Some More Criteria for Congruence of Triangles

Theorem 7.4 (SSS Congruence Rule)

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

Theorem 7.5 (RHS Congruence Rule)

If in two right triangles, the hypotenuse and one side are equal, the triangles are congruent.

Example 7:

Show that line PQ is the perpendicular bisector of AB given PA = PB and QA = QB.

Example 8:

Show that line AP bisects the angle between two lines l and m intersecting at A using P equidistant from l and m.

Exercise 7.3

Prove ∆ABD ≅ ∆ACD, ∆ABP ≅ ∆ACP and AP bisects both ∠A and ∠D; and is perpendicular bisector of BC.
Show AD bisects BC and ∠A given AD is an altitude in isosceles triangle ABC with AB = AC.
Prove ∆ABM ≅ ∆PQN and hence ∆ABC ≅ ∆PQR using equal medians and sides.
Use RHS rule to show triangle with equal altitudes is isosceles.
Show ∠B = ∠C by drawing AP ⊥ BC in isosceles triangle ABC.