Advertisements
Advertisements
प्रश्न
On the sides AB and AC of triangle ABC, equilateral triangle ABD and ACE are drawn. Prove that:
- ∠CAD = ∠BAE
- CD = BE
Advertisements
उत्तर
Given: ΔABD is an equilateral triangle.
ΔACE is an equilateral triangle
We need to prove that
(i) ∠CAD = ∠BAE
Proof:
(i) ΔABD is equilateral
∴ Each angle = 60°
⇒ ∠BAD = 60° ...(1)
Similarly,
ΔACE is equilateral
∴ Each angle = 60°
⇒ ∠CAE = 60° ...(2)
⇒ ∠BAD = ∠CAE ...[ from (1) and (2) ]...(3)
Adding ∠BAC to both sides, we have
⇒ ∠BAD + ∠BAC = ∠CAE + ∠BAC
⇒ ∠CAD = ∠BAE ...(4)
(ii) In ΔCAD and ΔBAE
AC = AE ...[ ΔACE is equilateral]
∠CAD = ∠BAE ... [ from (4) ]
AD = AB ...[ ΔABD is equilateral]
∴ By the Side-Angle-Side criterion of congruency,
ΔCAD ≅ ΔBAE
The corresponding parts of the congruent
triangles are congruent.
∴ CD = BE ...[ by c.p.c.t ]
Hence proved.
APPEARS IN
संबंधित प्रश्न
In an isosceles triangle, if the vertex angle is twice the sum of the base angles, then the measure of vertex angle of the triangle is
Which of the following is not a criterion for congruence of triangles?
Observe the information shown in pair of triangle given below. State the test by which the two triangles are congruent. Write the remaining congruent parts of the triangles.
From the information shown in the figure,
in ΔPTQ and ΔSTR
seg PT ≅ seg ST
∠PTQ ≅ ∠STR ...[Vertically opposite angles]
∴ ΔPTQ ≅ ΔSTR ...`square` test
∴ `{:("∠TPQ" ≅ square),("and" square ≅ "∠TRS"):}}` ...corresponding angles of congruent triangles
seg PQ ≅ `square` ...corresponding sides of congruent triangles
In the pair of triangles in the following figure, parts bearing identical marks are congruent. State the test and the correspondence of vertices by the triangle in pairs is congruent.
In ΔABC and ΔPQR and, AB = PQ, BC = QR and CB and RQ are extended to X and Y respectively and ∠ABX = ∠PQY. = Prove that ΔABC ≅ ΔPQR.
If the perpendicular bisector of the sides of a triangle PQR meet at I, then prove that the line joining from P, Q, R to I are equal.
ΔABC is isosceles with AB = AC. BD and CE are two medians of the triangle. Prove that BD = CE.
In triangles ABC and DEF, AB = FD and ∠A = ∠D. The two triangles will be congruent by SAS axiom if ______.
ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC.
The top and bottom faces of a kaleidoscope are congruent.
