Congruent Triangles: math test, Grade 10 – download the PDF

1 Angles of a triangle / 3 pts

In \(\triangle ABC\), \(m\angle A = (2x + 5)^\circ\), \(m\angle B = (3x - 5)^\circ\) and \(m\angle C = (x + 30)^\circ\).

1. Find \(x\) and the three angles.
2. Side \(\overline{AC}\) is extended past \(C\). Find the exterior angle at \(C\).

2 Which criterion? / 3 pts

For each pair of triangles \(\triangle ABC\) and \(\triangle DEF\), name the criterion that proves them congruent, or write “not enough”.

1. \(AB = DE\), \(\angle B = \angle E\), \(BC = EF\).
2. \(\angle A = \angle D\), \(\angle B = \angle E\), \(BC = EF\).
3. \(AB = DE\), \(BC = EF\), \(\angle A = \angle D\).

3 Hypotenuse-Leg / 3 pts

\(\triangle JKL\) is right at \(K\) and \(\triangle MNP\) is right at \(N\). \(JL = MP = 25\) in and \(JK = MN = 7\) in. Prove the triangles are congruent and find \(LK\) and \(PN\) (in inches and centimeters).

4 Isosceles triangle / 4 pts

In isosceles \(\triangle ABC\) with \(AB = AC\), the base angles are \(\angle B = (5y + 1)^\circ\) and \(\angle C = (3y + 17)^\circ\).

1. Find \(y\) and the base angles.
2. Find the apex angle \(\angle A\).

5 Proof in a parallelogram / 4 pts

In quadrilateral \(ABCD\), \(\overline{AB} \parallel \overline{DC}\) and \(AB = DC\). Prove that \(BC = DA\). Hint: draw diagonal \(\overline{AC}\).

6 Congruence by reflection / 3 pts

Let \(R(0, 0)\), \(S(6, 2)\), \(T(2, 5)\) and \(R'(0, 0)\), \(S'(2, 6)\), \(T'(5, 2)\). The second triangle is the reflection of the first across the line \(y = x\). Use squared distances to prove \(\triangle RST \cong \triangle R'S'T'\).