Question

cx is an altitude in triangle abc. which statements are true? select two options. △abc≅△bxc △axc∼△cxb △bcx≅△acx △acb∼△axc △cxa≅△cba

Explanation:

Step1: Recall the AA (Angle - Angle) similarity criterion

If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.
For \(\triangle AXC\) and \(\triangle CXB\):

• \(\angle AXC=\angle CXB = 90^{\circ}\) (given \(CX\) is an altitude)
• \(\angle A + \angle ACX=90^{\circ}\) and \(\angle ACX+\angle BCD = 90^{\circ}\), so \(\angle A=\angle BCD\) (by the property that in a right - triangle, the two non - right angles are complementary). So, \(\triangle AXC\sim\triangle CXB\) (by AA similarity)

Step2: For \(\triangle ACB\) and \(\triangle AXC\)

• \(\angle A=\angle A\) (common angle)
• \(\angle AXC=\angle ACB = 90^{\circ}\) (given \(CX\) is an altitude). So, \(\triangle ACB\sim\triangle AXC\) (by AA similarity)

Answer:

\(\triangle AXC\sim\triangle CXB\), \(\triangle ACB\sim\triangle AXC\)