Parallel and Perpendicular Lines: math practice, Grade 10 – download the PDF

1 Naming angle pairs ★★★

Use the figure, where \(a \parallel b\) and \(t\) is a transversal.

Name the type of each pair: (a) \(\angle 2\) and \(\angle 6\); (b) \(\angle 3\) and \(\angle 6\); (c) \(\angle 4\) and \(\angle 6\); (d) \(\angle 1\) and \(\angle 8\).

2 Finding angles from one measure ★★★

In the figure, \(a \parallel b\) and \(m\angle 3 = 65^\circ\). Find \(m\angle 6\), \(m\angle 7\), \(m\angle 2\) and \(m\angle 5\). Justify each answer.

3 Crossing lines ★★★

Two lines cross and one of the four angles measures \(38^\circ\). Find the other three angles.

4 True or false? ★★★

Is this statement true or false? “When a transversal crosses two lines, the corresponding angles are always congruent.” Explain.

5 Slope from two points ★★★

A line passes through \((0, 1)\) and \((4, 9)\). Find its slope, then the slope of (a) any parallel line and (b) any perpendicular line.

6 Parallel in disguise ★★★

Are the lines \(y = 4x - 5\) and \(8x - 2y = 7\) parallel? Explain.

7 Opposite reciprocals ★★★

Give the slope of a line perpendicular to a line with slope: (a) \(-\dfrac{5}{3}\); (b) \(4\); (c) \(\dfrac{1}{2}\).

8 Alternate interior angles and algebra ★★★

Two parallel lines are cut by a transversal. A pair of alternate interior angles measure \((4x + 7)^\circ\) and \((6x - 21)^\circ\). Find \(x\) and the two angle measures.

9 Same-side interior angles and algebra ★★★

Parallel lines are cut by a transversal. Same-side interior angles measure \((2x + 15)^\circ\) and \((3x + 25)^\circ\). Find \(x\) and both angles.

10 For which x are the lines parallel? ★★★

A transversal crosses lines \(p\) and \(q\). Two corresponding angles measure \((5x + 8)^\circ\) and \((7x - 12)^\circ\). For what value of \(x\) are \(p\) and \(q\) parallel? What is then the angle measure?

11 Parallel, perpendicular, or neither? ★★★

Line \(A\) passes through \((1, 2)\) and \((5, 5)\). Line \(B\) passes through \((0, 0)\) and \((3, -4)\). Line \(C\) passes through \((-2, 1)\) and \((2, 4)\). Classify each pair of lines: \(A\) and \(B\), \(A\) and \(C\).

12 A parallel line in standard form ★★★

Write an equation of the line through \((4, -3)\) that is parallel to \(2x + 5y = 10\).

13 A perpendicular line through a point ★★★

Find the equation of the line through \((6, 1)\) that is perpendicular to \(y = -3x + 2\).

14 The sprinkler pipe ★★★

A garden is drawn on a coordinate grid where 1 unit equals 1 meter. A water pipe runs along the line \(5x + 12y = 39\). A sprinkler is at the origin \((0, 0)\). What is the shortest distance from the sprinkler to the pipe, in meters and in feet (1 m \(\approx\) 3.28 ft)?

15 Perpendicular bisector equation ★★★

Find the equation of the perpendicular bisector of the segment with endpoints \(A(-2, 3)\) and \(B(4, 7)\).

16 Is the triangle a right triangle? ★★★

The vertices of triangle \(ABC\) are \(A(1, 1)\), \(B(5, 3)\) and \(C(3, 7)\).

(a) Show that the triangle has a right angle. (b) Find its area.

17 Is PQRS a rectangle? ★★★

Points \(P(0, 0)\), \(Q(4, 2)\), \(R(2, 6)\), \(S(-2, 4)\) are joined in order.

Use slopes to show that \(PQRS\) is a rectangle, then find its area.

18 Distance between parallel lines ★★★

Find the distance between the parallel lines \(y = 2x + 3\) and \(y = 2x - 7\). Give an exact value and a decimal rounded to hundredths.

19 Equidistant point on the x-axis ★★★

Find the point \(P\) on the \(x\)-axis that is equidistant from \(A(1, 4)\) and \(B(5, 2)\). Explain which theorem justifies why \(P\) lies on a particular line.

20 Two perpendiculars to a third line ★★★

In a plane, lines \(a\) and \(b\) are both perpendicular to a line \(c\). Prove that \(a \parallel b\) using angle pairs. Then explain the same result with slopes when \(c\) has slope \(\dfrac{2}{5}\).

21 Foot of the perpendicular ★★★

Let \(\ell\) be the line \(y = 2x - 4\) and \(P(0, 6)\). (a) Find the foot \(H\) of the perpendicular from \(P\) to \(\ell\). (b) Find \(PH\) and check it with the distance formula.