# Ex.7.2 Q4 Triangles Solution - NCERT Maths Class 9

## Question

\(ABC\) is a triangle in which altitudes \(BE\) and \(CF\) to sides \(AC\) and \(AB\) are equal (see the given figure).

Show that:

(i) \(\ \Delta ABE \cong \Delta ACF\)

(ii) \(AB=AC,\) i.e., ABC is an isosceles triangle.

## Text Solution

**What is Known?**

Altitudes,\(\text{BE}=\text{CF},\text{ BE}\bot \text{AC}\ \text{and CF}\bot \text{AB}\)

**To prove:**

(i) \(\ \Delta ABE \cong \Delta ACF\)

(ii) \(AB=AC,\) i.e., ABC is an isosceles triangle.

**Reasoning:**

We can show two triangles \(ABE\) and \(ACF\) congruent by using AAS congruency rule and then we can say corresponding parts of congruent triangles will be equal.

**Steps:**

(i) In \(\Delta ABE\text{ and }\Delta ACF,\)

\[\begin{align}& \angle AEB \text{ and }\angle AFC \;(\text{Each} 90^{\circ}) \\ \angle A&=\angle A \;(\text{Common angle}) \\ BE&=CF \; (\text{Given}) \\ \therefore \Delta ABE &\cong \Delta ACF \\&(\text{By AAS congruence rule}) \end{align}\]

(ii) It has already been proved that

\[\begin{align}&\Delta ABE \cong \Delta ACF, \\ &\therefore AB = AC \; (\text{By }CPCT) \\ \end{align}\]