Test: Properties of Isosceles Triangle - Class 9 MCQ

Angle acb = 75 ( linear pair) Ab= Ac hence angle B= angle c in triangle ABC angle a+B+c= 180. angle B+ c = 150 hence angle c a= 30

Since ST is parallel to QR and S is on PQ, by the Basic Proportionality Theorem (or Thales' theorem):

PS/SQ = PT/TR

However, since PQ=PR, triangles △PSQ and △PRT are similar due to the AA (Angle-Angle) criterion (one angle is common, and the other angles are corresponding angles formed by the parallel lines).

PS/SQ = PT/TR = 1

PS = PT

This is because in an equilateral triangle, all sides are equal, and the altitude divides the triangle into two 30-60-90 right triangles. In a 30-60-90 triangle, the ratio of the sides opposite to the angles 30°, 60°, and 90° are, respectively.

In an isosceles right-angled triangle:
- An isosceles triangle has two equal angles.
- A right-angled triangle has a 90° angle.
- In this case, the two acute angles are equal.
- So, each acute angle in an isosceles right-angled triangle will be 45°.
Therefore, the correct answer is C: 45°, 45°.

In triangle ABC with AB = BC:
- Angles A and C are equal (congruent) because the sides opposite them are equal in an isosceles triangle.
- So, the correct answer is:
- Angle A = Angle C (Option C)

The altitudes of a triangle are the line segments drawn from each side of the triangle to the angle opposite that side, such that the line segment is perpendicular to the side it is drawn from. In an isosceles triangle, only the altitudes of the legs of equal length in the triangle are congruent. 

By using the theorem,If two sides of a triangle are equal then the opposite angles to the sides are equal.
⇒ if PQ=PR then ∠Q=∠R
in triangle PQR,
⇒ ∠P+∠Q+∠R=180°
⇒ ∠P+∠Q+∠Q=180° (∵∠Q=∠R)
⇒ ∠P+65°+65°=180°
⇒ ∠P+130°=180°
⇒ ∠P=180°-130°
⇒ ∠P=50