These 26 problems cover Common Core standards G-CO.C.10 and G-C.A.3: the Triangle Angle Sum and Exterior Angle theorems, the Isosceles Triangle Theorem, midsegments, medians and the centroid, and the circumcenter, incenter, and orthocenter with their circles.
The first problems use angle facts and algebra. Later ones use coordinates and circles, and the last six are word problems and multi-step tasks, so show every step.
Problem 1: Find the third angle
In \(\triangle ABC\), \(m\angle A=52^{\circ}\) and \(m\angle B=67^{\circ}\). Find \(m\angle C\).
Problem 2: Acute angles of a right triangle
In right triangle ABC, \(m\angle C=90^{\circ}\) and \(m\angle A=35^{\circ}\). Find \(m\angle B\).
Problem 3: Base angles of an isosceles triangle
Triangle ABC is isosceles with \(\overline{AB}\cong\overline{AC}\) and a vertex angle of \(m\angle A=40^{\circ}\). Find the measure of each base angle.
Problem 4: Angle sum with variables
The angles of a triangle measure \((x+10)^{\circ}\), \((2x-5)^{\circ}\), and \((3x+7)^{\circ}\). Find \(x\) and all three angle measures.
Problem 5: Use the exterior angle
In the figure, side \(\overline{BC}\) is extended past C. Find the measure of the exterior angle at C.
Problem 6: Missing remote angle
An exterior angle of a triangle measures \(128^{\circ}\). One remote interior angle measures \(75^{\circ}\). Find the other remote interior angle.
Problem 7: Exterior angle with variables
An exterior angle measures \((6x-4)^{\circ}\). The remote interior angles measure \((2x+10)^{\circ}\) and \((3x+6)^{\circ}\). Find \(x\) and the exterior angle.
Problem 8: Isosceles triangle with variables
In isosceles triangle ABC, \(\overline{AB}\cong\overline{AC}\), \(m\angle B=(3x+4)^{\circ}\), and \(m\angle C=(5x-16)^{\circ}\). Find \(x\), the base angles, and \(m\angle A\).
Problem 9: Equal angles, equal sides
In \(\triangle PQR\), \(m\angle P=m\angle Q=58^{\circ}\), \(PR=(2x+1)\text{ cm}\), and \(QR=(4x-13)\text{ cm}\). Find \(x\), \(PR\), and \(m\angle R\).
Problem 10: Midsegment length
In \(\triangle ABC\), D and E are the midpoints of \(\overline{AB}\) and \(\overline{AC}\), and \(BC=26\). Find \(DE\).
Problem 11: Midsegment with variables
In \(\triangle ABC\), \(\overline{DE}\) is the midsegment parallel to \(\overline{BC}\). \(DE=(4x-3)\text{ in}\) and \(BC=(7x+2)\text{ in}\). Find \(x\), \(DE\), and \(BC\).
Problem 12: Midsegment on a grid
Triangle ABC has \(A(2,8)\), \(B(-2,0)\), and \(C(10,0)\). D is the midpoint of \(\overline{AB}\) and E is the midpoint of \(\overline{AC}\). Find the coordinates of D and E, then show that \(\overline{DE}\parallel\overline{BC}\) and \(DE=\frac{1}{2}BC\).
Problem 13: Segments of a median
The medians \(\overline{AD}\), \(\overline{BE}\), and \(\overline{CF}\) of \(\triangle ABC\) meet at the centroid G. If \(AD=21\), find \(AG\) and \(GD\).
Problem 14: Centroid with variables
G is the centroid of \(\triangle ABC\) and D is the midpoint of \(\overline{AC}\), so \(\overline{BD}\) is a median. \(BG=(5x-1)\text{ in}\) and \(GD=(2x+1)\text{ in}\). Find \(x\), \(BG\), and \(BD\).
Problem 15: Centroid on a grid
Find the centroid of the triangle with vertices \(A(1,2)\), \(B(7,2)\), and \(C(4,8)\).
Problem 16: Name the center
Name the point of concurrency described: (a) the intersection of the three medians; (b) the intersection of the perpendicular bisectors of the sides; (c) the intersection of the angle bisectors; (d) the intersection of the lines containing the altitudes.
Problem 17: Circumcenter of a right triangle
Right triangle ABC has \(A(0,0)\), \(B(12,0)\), and \(C(0,5)\). Find the circumcenter and the radius of the circumscribed circle.
Problem 18: Circumcenter by perpendicular bisectors
Triangle ABC has \(A(0,0)\), \(B(6,0)\), and \(C(2,6)\). Find the circumcenter by writing the perpendicular bisectors of \(\overline{AB}\) and \(\overline{AC}\).
Problem 19: Incenter and inscribed circle
Right triangle ABC has legs of 5 ft and 12 ft and a hypotenuse of 13 ft. The incenter I is the center of the inscribed circle. (a) Find the area and the semiperimeter \(s\). (b) Use \(\text{area}=r\cdot s\) to find the radius \(r\) of the inscribed circle. (c) How far is I from each side?
Problem 20: Angles at the incenter
In \(\triangle ABC\), \(m\angle A=50^{\circ}\) and \(m\angle B=70^{\circ}\). I is the incenter. Find \(m\angle BAI\), \(m\angle ABI\), and \(m\angle AIB\).
Problem 21: Where to put the fountain
A park designer puts a fountain exactly the same distance from three trees. On a grid in yards, the Oak is at \((0,0)\), the Elm is at \((12,0)\), and the Pine is at \((4,8)\). Find the fountain’s location and its distance from each tree to the nearest tenth of a yard.
Problem 22: Largest circle in a triangle
Diego cuts the largest possible circle from a triangular piece of plywood with sides of 9 ft, 12 ft, and 15 ft. What is the diameter of the circle?
Problem 23: Balance point of a triangle
Priya cuts a triangle from uniform cardboard with vertices at \((0,0)\), \((12,0)\), and \((3,9)\) (inches). She wants to balance it on a pencil tip. Where should the tip touch the triangle?
Problem 24: Roof gable angles
The end of a roof is an isosceles triangle ABC with \(\overline{AC}\cong\overline{CB}\) and a peak angle of \(m\angle C=110^{\circ}\). The base \(\overline{AB}\) is extended past B. (a) Find the base angles. (b) Find the exterior angle at B, and check it with the Exterior Angle Theorem.
Problem 25: Midsegment triangle perimeter
Triangle ABC has sides \(AB=14\text{ in}\), \(BC=22\text{ in}\), and \(CA=18\text{ in}\). D, E, and F are the midpoints of \(\overline{AB}\), \(\overline{BC}\), and \(\overline{CA}\). Find the three side lengths and the perimeter of \(\triangle DEF\), and compare it with the perimeter of \(\triangle ABC\).
Problem 26: Orthocenter on a grid
Find the orthocenter of the triangle with \(A(0,0)\), \(B(8,0)\), and \(C(2,6)\). Check your answer with the third altitude.
See every chapter on the Geometry math page.