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

1 Third angle ★★★

Two angles of a triangle measure \(52^\circ\) and \(71^\circ\). Find the third angle.

2 An exterior angle ★★★

The two remote interior angles of a triangle measure \(38^\circ\) and \(64^\circ\). Find the exterior angle, then check with the adjacent interior angle.

3 True or false? ★★★

Decide whether each statement is true or false and justify.

1. Two sides and a non-included angle (SSA) prove two triangles congruent.
2. Two angles and a non-included side (AAS) prove two triangles congruent.
3. If three angles match, the triangles are congruent.
4. Congruent triangles have equal areas.

4 Corresponding parts ★★★

\(\triangle PQR \cong \triangle XYZ\). Given \(PQ = 9\) cm, \(QR = 12\) cm and \(m\angle R = 41^\circ\), find \(XY\), \(YZ\) and \(m\angle Z\).

5 Name the criterion ★★★

Name the criterion (SSS, SAS, ASA, AAS, HL, or none) that proves the triangles congruent.

1. Two sides and the angle between them.
2. Two angles and the side between them.
3. Two angles and a side not between them.
4. Three pairs of equal sides.
5. Two sides and an angle not between them.
6. Two right triangles with equal hypotenuses and one pair of equal legs.

6 Isosceles and equilateral ★★★

1. An isosceles triangle has an apex angle of \(38^\circ\). Find each base angle.
2. An equilateral triangular banner has a perimeter of 45 in. Find the side length (in inches and centimeters) and each angle.

7 Angles with algebra ★★★

The angles of a triangle measure \((x + 10)^\circ\), \((2x - 20)^\circ\) and \((3x + 10)^\circ\). Find \(x\) and the three angles. What kind of triangle is it?

8 Solve for x ★★★

\(\triangle ABC \cong \triangle DEF\).

1. \(AB = (3x + 2)\) cm and \(DE = (5x - 10)\) cm. Find \(x\) and \(AB\).
2. \(m\angle B = (2y + 9)^\circ\) and \(m\angle E = (3y - 11)^\circ\). Find \(y\) and \(m\angle B\).

9 Exterior angle equation ★★★

An exterior angle of a triangle measures \((7x - 5)^\circ\). The remote interior angles measure \((2x + 20)^\circ\) and \((3x + 15)^\circ\). Find \(x\), the three angles involved, and the adjacent interior angle.

10 Ladders and HL ★★★

Two ladders lean against vertical walls on level ground. \(\triangle ABC\) is right at \(B\) and \(\triangle DEF\) is right at \(E\). Both ladders (the hypotenuses \(AC\) and \(DF\)) are 17 ft long, and both reach a point 8 ft up the wall (\(AB = DE = 8\) ft). Show that the triangles are congruent and find how far each foot is from its wall.

11 Base angles with algebra ★★★

In isosceles \(\triangle ABC\) with \(AB = AC\), the base angles are \(\angle B = (4x - 6)^\circ\) and \(\angle C = (2x + 30)^\circ\). Find \(x\), then all three angles.

12 Why SSA fails ★★★

In \(\triangle ABC\), \(m\angle A = 30^\circ\), \(AB = 8\) and \(BC = 5\). Show that two different triangles fit this data by finding the two possible lengths of \(AC\) to the nearest hundredth. Hint: the height from \(B\) to line \(AC\) is \(8 \sin 30^\circ\).

13 The kite ★★★

In quadrilateral \(ABCD\), \(AB = AD\) and \(CB = CD\). Prove that \(\angle B = \angle D\).

14 Congruent on a grid ★★★

The vertices are \(A(-3, 1)\), \(B(1, 1)\), \(C(-3, 4)\) and \(D(2, -2)\), \(E(5, -2)\), \(F(2, 2)\). Are \(\triangle ABC\) and \(\triangle DEF\) congruent? If so, write the congruence with vertices in matching order.

15 Points on an angle bisector ★★★

Ray \(BP\) bisects \(\angle ABC\). Point \(P\) lies on the bisector, \(\overline{PD} \perp \overline{BA}\) at \(D\) and \(\overline{PE} \perp \overline{BC}\) at \(E\). Prove that \(PD = PE\).

16 The median of an isosceles triangle ★★★

In \(\triangle ABC\), \(AB = AC\) and \(M\) is the midpoint of \(\overline{BC}\). Prove that \(\angle B = \angle C\) and that \(\overline{AM} \perp \overline{BC}\).

17 The converse ★★★

In \(\triangle ABC\), \(\angle B = \angle C\). Let \(\overline{AD}\) bisect \(\angle BAC\) with \(D\) on \(\overline{BC}\). Prove that \(AB = AC\).

18 Equal distances in an equilateral triangle ★★★

\(\triangle ABC\) is equilateral. Point \(D\) is on \(\overline{AB}\) and point \(E\) is on \(\overline{BC}\) with \(AD = BE\). Prove that \(CD = AE\).

19 A rotation on the grid ★★★

Let \(A(1, 1)\), \(B(7, 3)\), \(C(3, 6)\) and \(A'(-1, 1)\), \(B'(-3, 7)\), \(C'(-6, 3)\).

1. Show that \(\triangle ABC \cong \triangle A'B'C'\) using squared distances.
2. Describe the rigid motion that maps one onto the other.

20 Across the pond ★★★

Points \(A\) and \(B\) are on opposite shores of a pond. A surveyor stands at a point \(C\) from which both are visible. She extends \(\overline{AC}\) past \(C\) to a point \(D\) with \(CD = CA\), and extends \(\overline{BC}\) past \(C\) to a point \(E\) with \(CE = CB\). She measures \(DE = 46\) m. Explain why the pond measures \(AB = 46\) m and convert to feet.

21 Exterior angle and a ratio ★★★

An exterior angle at vertex \(C\) of \(\triangle ABC\) measures \(130^\circ\). The remote interior angles satisfy \(m\angle A : m\angle B = 3 : 2\). Find all three interior angles of the triangle.

22 Equal altitudes ★★★

In \(\triangle ABC\), \(AB = AC\). The altitude from \(B\) meets \(\overline{AC}\) at \(E\) and the altitude from \(C\) meets \(\overline{AB}\) at \(F\). Prove that \(BE = CF\).