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Triangles

Short Answer Type

31. ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see figure). Show that:

(i) ∆ABE ≅ ∆ACF
(ii) AB = AC, i.e., ∆ABC is an isosceles triangle.

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32. ABC and DBC are two isosceles triangles on the same base BC (see figure). Show that ∠ABD = ∠ACD.

1318 Views

Long Answer Type

33.
∆ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see figure). Show that ∆BCD is a right angle.

Given: ∆ABC is an isosceles triangle in which AB = AC.
Side BA is produced to D such that AD = AB.
To Prove: ∠BCD is a right angle.
Proof: ∵ ABC is an isosceles triangle
∴ ∠ABC = ∠ACB ...(1)
∵ AB = AC and AD = AB
∴ AC = AD

∴ In ∆ACD,
∠CDA = ∠ACD
| Angles opposite to equal sides of a triangle are equal
⇒ ∠CDB = ∠ACD    ...(2)
Adding the corresponding sides of (1) and (2), we get
∠ABC + ∠CDB = ∠ACB + ∠ACD
⇒ ∠ABC + ∠CDB = ∠BCD    ...(3)
In ∆BCD,
∠BCD + ∠DBC + ∠CDB = 180°
| ∵ Sum of all the angles of a triangle is 180°
⇒ ∠BCD + ∠ABC + ∠CDB = 180°
⇒    ∠BCD + ∠BCD = 180°
| Using (3)
⇒    2∠BCD = 180°
⇒    ∠BCD = 90°
⇒ ∠BCD is a right angle.

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Short Answer Type

34. ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.

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35. Show that the angles of an equilateral triangle are 60° each.

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36. Angles A, B and C of a triangle ABC are equal to each other. Prove that ∆ABC is equilateral.

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37. BD and CE are the bisectors of ∠B and ∠C of an isosceles triangle ABC with AB = AC. Prove that BD = CE.

1965 Views

Long Answer Type

38.

ABC is a triangle in which ∠B = 2∠C. D is a point on side BC such that AD bisects ∠BAC and AB = CD. Prove that ∠BAC = 72°.
[Hint. Take a point P on AC such that BP bisects ∠B. Join P and D.]

1214 Views