7.1-Congruence of Triangles

7.1-Congruence of Triangles Important Formulae

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Grade 9 → Math → Triangles → 7.1-Congruence of Triangles

After successful completion of this topic, you should be able to:

  • Understand 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 two sides and the included angle of the other triangle.
Theorem 7.1 (ASA congruence rule) : Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.


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In quadrilateral ACBD, AC = AD and AB bisects ∠A (see Fig. 7.16). Show that ∆ ABC ≅ ∆ ABD. What can you say about BC and BD?

Solution:
They are equal.

Suggest improvements/Report errors

ABCD is a quadrilateral in which AD = BC and ∠ DAB = ∠ CBA (see Fig. 7.17). Prove that:

(i) ∆ ABD ≅ ∆ BAC
(ii) BD = AC
(iii) ∠ABD=∠BAC.

AD and BC are equal perpendiculars to a line segment AB (see Fig. 7.18). Show that CD bisects AB.

l and m are two parallel lines intersected by another pair of parallel lines p and q (see Fig. 7.19). Show that ∆ ABC ≅ ∆ CDA.

Line l is the bisector of an angle ∠A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠ A (see Fig. 7.20). Show that:
(i) ∆APB ≅ ∆AQB
(ii) BP = BQ or B is equidistant from the arms of ∠A.

In Fig. 7.21,AC = AE, AB = AD and ∠BAD =∠EAC. Show that BC = DE.

AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that ∠ BAD = ∠ ABE and ∠ EPA = ∠ DPB (see Fig. 7.22). Show that:
(i) ∆DAP ≅ ∆EBP
(ii) AD = BE

In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see Fig. 7.23). Show that:
(i) ∆ AMC ≅ ∆ BMD
(ii) ∠DBC is a right angle.
(iii) ∆ DBC ≅ ∆ ACB
(iv) CM = $\dfrac{1}{2}$ AB