In quadrilateral ACBD, AC = AD, and AB bisect ∠A (see Fig. 7.16). Show that ΔABC ≅ ΔABD.
What can you say about BC and BD?
ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Fig. 7.17). Prove that
(i) ΔABD ≅ ΔBAC
(ii) BD = AC
(iii) ∠ABD = ∠BAC.
l and m are two parallel lines intersected by another pair of parallel lines p and q (see Fig. 7.19). Show that ΔABC ≅ ΔCDA.
In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other
at O. Join A to O. Show that :
(i) OB = OC
(ii) AO bisects ∠A
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see Fig. 7.32).
Show that
(i) ΔABE ≅ ΔACF
(ii) AB = AC, i.e., ABC is an isosceles triangle.
ABC is a right-angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.
AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that
(i) AD bisects BC
(ii) AD bisects ∠A.
Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see Fig. 7.40). Show that:
(i) ΔABM ≅ ΔPQN
(ii) ΔABC ≅ ΔPQR
ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that ∠B = ∠C.
Show that of all line segments drawn from a given point, not on it, the perpendicular line segment is the shortest.
ABC is a triangle. Locate a point in the interior of ΔABC which is equidistant from all the vertices of ΔABC.
AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (see Fig.7.50).
Show that ∠A > ∠C and ∠B > ∠D.
Complete the hexagonal and star-shaped Rangolies [see Fig. (i) and (ii)] by filling them with as many equilateral triangles of side 1 cm as you can. Count the number of triangles in each case. Which has more triangles?
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