These 23 problems practice the precise definitions of geometry (G-CO.A.1). Students name points and rays, use the Segment Addition and Angle Addition Postulates, work with midpoints and bisectors, and solve for unknowns in complementary, supplementary, linear and vertical angle pairs.
The problems go from naming and definitions to algebra and word problems about a highway, a ladder and a laptop. Draw each figure before solving.
Problem 1: Naming points and rays
Use the figure for the following.
- Name three collinear points.
- Name three noncollinear points.
- Name the ray with endpoint Q that passes through R.
Problem 2: Defining a segment
Write a precise definition of a line segment using the undefined terms point and line.
Problem 3: Segment addition: find the whole
Point B is between A and C on a straight board. Find AC.
Problem 4: Segment addition: find a part
Point B is between A and C. AC = 31 cm and AB = 18 cm. Find BC.
Problem 5: Segment addition with algebra
Point B is between A and C, with \(AB=3x+2\), \(BC=2x+7\) and \(AC=34\). Find x, AB and BC.
Problem 6: Midpoint with algebra
M is the midpoint of \(\overline{PQ}\), with \(PM=4x-3\) and \(MQ=2x+9\). Find x and PQ.
Problem 7: Midpoint on a number line
Point A is at \(-4\) and point B is at \(10\) on a number line. Find AB and the coordinate of the midpoint of \(\overline{AB}\).
Problem 8: Angle addition: find the whole
Ray BD lies inside \(\angle ABC\). Find \(m\angle ABC\).
Problem 9: Angle addition: find a part
Ray OB lies inside \(\angle AOC\). \(m\angle AOC\)\(=112^{\circ}\) and \(m\angle AOB\)\(=65^{\circ}\). Find \(m\angle BOC\).
Problem 10: Angle addition with algebra
Ray QS lies inside \(\angle PQR\). \(m\angle PQR\)\(=96^{\circ}\), \(m\angle PQS\)\(=(4x+8)^{\circ}\) and \(m\angle SQR\)\(=(2x+4)^{\circ}\). Find x, \(m\angle PQS\) and \(m\angle SQR\).
Problem 11: Angle bisector
Ray BD bisects \(\angle ABC\). \(m\angle ABD\)\(=(3x+5)^{\circ}\) and \(m\angle DBC\)\(=(5x-9)^{\circ}\). Find x and \(m\angle ABC\).
Problem 12: Complement of an angle
Find the complement of an angle that measures \(63^{\circ}\).
Problem 13: Supplement of an angle
Find the supplement of an angle that measures \(127^{\circ}\).
Problem 14: Complementary angles with algebra
Two angles are complementary. Their measures are \((2x+3)^{\circ}\) and \((3x+7)^{\circ}\). Find x and both angle measures.
Problem 15: Supplementary angles with algebra
Two angles are supplementary. Their measures are \((4x-10)^{\circ}\) and \((6x+20)^{\circ}\). Find x and both angle measures.
Problem 16: Angles at an intersection
Two lines intersect. One of the four angles measures \(48^{\circ}\). Find the measures of angles a, b and c.
Problem 17: Vertical angles with algebra
Angles 1 and 3 are vertical angles. \(m\angle 1\)\(=(7x-12)^{\circ}\) and \(m\angle 3\)\(=(4x+18)^{\circ}\). Find x, \(m\angle 1\) and \(m\angle 2\).
Problem 18: Perpendicular lines
Lines s and t are perpendicular. One of the angles they form measures \((3x+6)^{\circ}\). Find x.
Problem 19: Points and a circle
A circle has center C and radius 6 cm. Point D is 6 cm from C, point E is 4 cm from C and point F is 8 cm from C, all in the same plane. Which points are on the circle?
Problem 20: Road trip on a straight highway
A diner is at mile marker 18 and a motel is at mile marker 74 on a straight highway. Jamal wants a rest stop at the midpoint between them. Find the distance between the diner and the motel, and the mile marker of the rest stop.
Problem 21: Ladder against a wall
A ladder leans against a vertical wall on level ground. It makes a \(75^{\circ}\) angle with the ground. Find the angle between the ladder and the wall.
Problem 22: Laptop lid
Maya opens her laptop so the lid makes a \(115^{\circ}\) angle with the keyboard. She lowers the lid by \(40^{\circ}\). Find the new angle between the lid and the keyboard.
Problem 23: Four angles at an intersection
Two lines intersect to form angles 1, 2, 3 and 4 in order around the point. Angles 1 and 2 form a linear pair, with \(m\angle 1\)\(=(2x+15)^{\circ}\) and \(m\angle 2\)\(=(3x-5)^{\circ}\). Find x and the measures of all four angles.
See every chapter on the Geometry math page.