Question

prove: \\( \triangle hkj \sim \triangle lnp \\)
100\\( ^{\circ} \\)

1. \\( m\angle h + m\angle j + m\angle k = 180^{\circ} \\) 2.?
2. \\( 30^{\circ} + 50^{\circ} + m\angle k = 180^{\circ} \\) 3. substitution property
3. \\( 80^{\circ} + m\angle k = 180^{\circ} \\) 4. addition
4. \\( m\angle k = 100^{\circ} \\) 5. subtraction property of equality
5. \\( m\angle j = m\angle p; m\angle k = m\angle n \\) 6. substitution
6. \\( \angle j \cong \angle p; \angle k \cong \angle n \\) 7. if angles are equal then they are congruent
7. \\( \triangle hkj \sim \triangle lnp \\) 8. aa similarity theorem

which reason is missing in step 2?
cpctc
definition of supplementary angles

Explanation:

In a triangle, the sum of the interior angles is always \(180^{\circ}\). This is a fundamental property of triangles in geometry. Step 2 in the proof is stating that the sum of the angles of \(\triangle HKJ\) (\(m\angle H + m\angle J + m\angle K\)) equals \(180^{\circ}\), which is based on the triangle angle - sum theorem.

CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is used when dealing with congruent triangles, not for the sum of angles in a single triangle. The definition of supplementary angles is about two angles that add up to \(180^{\circ}\), not three angles in a triangle.

Answer:

Triangle angle - sum theorem.