Question

cx is an altitude in triangle abc.
which statements are true? choose two correct answers.
□ $\triangle axc \sim \triangle cxb$
□ $\triangle abc \cong \triangle bxc$
□ $\triangle cxa \cong \triangle cba$
□ $\triangle bcx \cong \triangle acx$
□ $\triangle acb \sim \triangle axc$

Explanation:

Step1: Analyze \(\triangle AXC \sim \triangle CXB\)

Since \(CX\) is an altitude, \(\angle AXC=\angle CXB = 90^\circ\). Also, \(\angle A=\angle A\) (common angle for \(\triangle AXC\) and \(\triangle ACB\), but for \(\triangle AXC\) and \(\triangle CXB\), \(\angle A+\angle ACX = 90^\circ\) and \(\angle ACX+\angle BCX=90^\circ\), so \(\angle A=\angle BCX\). By AA similarity, \(\triangle AXC\sim\triangle CXB\).

Step2: Analyze \(\triangle ACB \sim \triangle AXC\)

\(\angle AXC = \angle ACB=90^\circ\) (since \(CX\) is altitude and \(\angle ACB\) is right angle? Wait, no, the diagram shows \(\angle C\) as right angle? Wait, the diagram has \(\angle C\) with a right angle and \(CX\) as altitude. So \(\angle A\) is common to \(\triangle ACB\) and \(\triangle AXC\), and \(\angle AXC=\angle ACB = 90^\circ\). By AA similarity, \(\triangle ACB\sim\triangle AXC\).

Now check other options: \(\triangle ABC\cong\triangle BXC\) is false (different sizes), \(\triangle CXA\cong\triangle CBA\) is false (congruence requires equal sides, not just angles), \(\triangle BCX\cong\triangle ACX\) is false (unless \(AC = BC\), not given). So the true ones are \(\triangle AXC \sim \triangle CXB\) and \(\triangle ACB \sim \triangle AXC\) (i.e., \(\triangle ACB \sim \triangle AXC\) is same as \(\triangle ACB \sim \triangle AXC\) or \(\triangle AXC \sim \triangle ACB\)).

Answer:

The two correct answers are:

• \(\boldsymbol{\triangle AXC \sim \triangle CXB}\)
• \(\boldsymbol{\triangle ACB \sim \triangle AXC}\) (or \(\boldsymbol{\triangle ACB \sim \triangle AXC}\) which is the last option \(\triangle ACB \sim \triangle AXC\))