Triangle Theorems and Points of Concurrency: Geometry Math Test – PDF Download

Question 1: Angle sum with variables (2 points)

The angles of a triangle measure \((2x+4)^{\circ}\), \((3x-1)^{\circ}\), and \((x+15)^{\circ}\).

1. Find \(x\). (1 point)
2. Find the measure of the largest angle. (1 point)

Question 2: Exterior angle equation (2 points)

The figure shows a triangle with one side extended. The remote interior angles measure \((3x+8)^{\circ}\) and \((2x+16)^{\circ}\), and the exterior angle measures \((7x-4)^{\circ}\).

1. Find \(x\). (1 point)
2. Find the measure of the exterior angle. (1 point)

Question 3: Isosceles triangle angles (2 points)

In \(\triangle ABC\), \(\overline{AB}\cong\overline{AC}\), \(m\angle B=(4x+6)^{\circ}\), and \(m\angle C=(6x-14)^{\circ}\).

1. Find \(x\) and \(m\angle B\). (1 point)
2. Find \(m\angle A\). (1 point)

Question 4: Midsegment theorem (2 points)

In \(\triangle ABC\), D and E are the midpoints of \(\overline{AB}\) and \(\overline{AC}\).

1. Find \(x\), \(DE\), and \(BC\). (1 point)
2. Find the perimeter of \(\triangle ADE\). (1 point)

Question 5: Centroid segments (2 points)

G is the centroid of \(\triangle ABC\), and \(\overline{BE}\) is a median with \(BG=(3x+5)\text{ in}\) and \(GE=(x+4)\text{ in}\).

1. Find \(x\). (1 point)
2. Find \(BE\). (1 point)

Question 6: Name the center (2 points)

For each description, name the center of the triangle.

1. The center of the largest circle that fits inside the triangle and touches all three sides. (1 point)
2. The center of the circle that passes through all three vertices. (1 point)

Question 7: Centers on the plane (3 points)

Triangle ABC has vertices \(A(0,0)\), \(B(12,0)\), and \(C(0,9)\).

1. Find the circumcenter and the radius of the circumscribed circle. (2 points)
2. Find the centroid. (1 point)

Question 8: Prove the angle sum (3 points)

Given: \(\triangle ABC\). Line \(DE\) passes through C and is parallel to \(\overline{AB}\), with D on the left of C and E on the right. Prove \(m\angle A+m\angle B+m\angle ACB=180^{\circ}\). Write a two-column proof.

Question 9: Incenter angles (2 points)

I is the incenter of \(\triangle ABC\), \(m\angle A=64^{\circ}\), and \(m\angle B=48^{\circ}\).

1. Find \(m\angle IAB\) and \(m\angle IBA\). (1 point)
2. Find \(m\angle AIB\). (1 point)

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