All questions of Triangles for Class 9 Exam

Understanding the Geometry of Point P
When a point P is equidistant from two intersecting lines l and m at point A, it creates specific angles that relate to point P and the lines.
Key Concepts
- Equidistant Definition: Point P being equidistant from lines l and m means that the perpendicular distances from P to both lines are equal.
- Intersection at Point A: The lines l and m intersect at point A, creating angles around A.
Angles Formed by Point P
- Angle Relationships: The angles formed at point A can be expressed in terms of P:
- Angle BAP is the angle between line l and line segment AP.
- Angle CAP is the angle between line m and line segment AP.
- Angle BPA is the angle between line l and line segment BP.
Reasoning for the Correct Answer
- Since P is equidistant from lines l and m, the angles formed with respect to A will have specific relationships:
- Angle CAP = Angle BAP: This is true because both angles are subtended by the same line segment AP from point P, making them equal.
- Angle CAP does not equal Angle BPA: These angles are not directly related as they involve different line segments and points.
Conclusion
Thus, the correct answer is option 'C':
Angle CAP = Angle BAP.
This equality arises due to the symmetry created by point P being equidistant from both lines.

If an exterior angle of a triangle is 100 degrees, then the sum of the two interior angles adjacent to it is 180 - 100 = 80 degrees.

Understanding Triangle Properties
In triangle ABC, we are given the angles:
- Angle A = 50°
- Angle B = 60°
To determine the longest side, we need to utilize the property that states: the side opposite the largest angle is the longest side.
Calculating Angle C
First, we calculate the third angle (C) using the triangle angle sum property:
- Angle C = 180° - (Angle A + Angle B)
- Angle C = 180° - (50° + 60°) = 70°
Now we have all angles in triangle ABC:
- Angle A = 50°
- Angle B = 60°
- Angle C = 70°
Identifying the Longest Side
Next, we compare the angles to determine which is the largest:
- Angle A = 50°
- Angle B = 60°
- Angle C = 70° (Largest angle)
The side opposite the largest angle (Angle C) is side AB.
Conclusion
Since side AB is opposite the largest angle (70°), it is the longest side of triangle ABC.
Thus, the correct answer is:
- a) AB
This aligns with the triangle inequality property, confirming that side AB is indeed the longest side in triangle ABC.

Answer:

Explanation:
In a triangle, the sum of all angles is always 180 degrees. Let's assume the three angles of the triangle are A, B, and C.

Given:
One angle (let's assume A) is equal to the sum of the other two angles (B and C).

To prove:
The triangle is a right-angled triangle.

Proof:
Let's assume that angle A is equal to the sum of angles B and C.
So, A = B + C

Now, let's assume that the triangle is not a right-angled triangle. In this case, the sum of angles B and C would be less than 90 degrees (acute-angled triangle) or greater than 90 degrees (obtuse-angled triangle).

Case 1: Acute-angled triangle
If the sum of angles B and C is less than 90 degrees, it means that angle A (which is equal to B + C) would also be less than 90 degrees. However, this contradicts the given condition that angle A is equal to the sum of angles B and C. Therefore, an acute-angled triangle cannot satisfy the given condition.

Case 2: Obtuse-angled triangle
If the sum of angles B and C is greater than 90 degrees, it means that angle A (which is equal to B + C) would also be greater than 90 degrees. However, this again contradicts the given condition that angle A is equal to the sum of angles B and C. Therefore, an obtuse-angled triangle cannot satisfy the given condition.

Hence, the only possibility left is that the triangle is a right-angled triangle, where the sum of angles B and C is equal to 90 degrees. In this case, angle A (which is equal to B + C) would also be equal to 90 degrees.

Therefore, the correct answer is option 'D' - Right-angled triangle.

Explanation:

To understand why the line joining the vertices of two isosceles triangles with a common base bisects them at a right angle, let's consider the properties of isosceles triangles.

Properties of Isosceles Triangles:
1. In an isosceles triangle, the two sides opposite the equal angles are of equal length.
2. The angles opposite the equal sides of an isosceles triangle are equal.

Given:
We have two isosceles triangles with a common base.

To Prove:
The line joining their vertices bisects the triangles at a right angle.

Proof:
Let's label the triangles as ABC and ABD, where AB is the common base and AC = BC and AD = BD.

Now, let's draw the line joining the vertices C and D.

Step 1:
Since AC = BC and AD = BD, we can conclude that triangles ABC and ABD are congruent by the Side-Side-Side (SSS) congruence criterion.

Step 2:
Using the congruence of the triangles, we can say that angle BAC is congruent to angle BAD and angle ABC is congruent to angle ABD.

Step 3:
Since angles BAC and BAD are congruent, and angles ABC and ABD are congruent, we can conclude that angle CAB is congruent to angle DAB.

Step 4:
When two angles are congruent, they are called vertically opposite angles. Vertically opposite angles are always equal.

Step 5:
Since angle CAB and angle DAB are vertically opposite angles and congruent, we can conclude that the line joining the vertices C and D bisects the triangles ABC and ABD at a right angle.

Hence, the correct answer is option 'C' - The line joining the vertices of two isosceles triangles with a common base bisects them at a right angle.