Praxis Geometry (5163)

Expect these ideas to appear on their own as quick computation items and, just as often, as one step inside a longer problem about congruence, similarity, coordinates, or circles. The four chapters move from lines and angles, to triangles, to quadrilaterals and other polygons, and finally to perimeter and area. Work each worked example yourself before reading the steps, because the test rewards fast, accurate setup more than memorized facts.

This chapter covers outline area I.B, properties of lines and angles: parallel, perpendicular, and intersecting lines; supplementary, complementary, and vertical angles; linear pairs; and the angle pairs formed when a transversal crosses two lines. These relationships drive a large share of the algebra-in-geometry questions on the test, and they also supply the reasons you will cite in proofs in Section 2.

A line extends without end in both directions and is determined by any two of its points. A ray has one endpoint and extends forever in one direction; ray AB starts at A and passes through B. A segment has two endpoints. An angle is formed by two rays with a common endpoint, called the vertex. Angles are classified by measure: acute (between 0° and 90°), right (exactly 90°), obtuse (between 90° and 180°), and straight (exactly 180°).

Two lines in a plane either intersect in exactly one point or are parallel (never meet). Perpendicular lines intersect to form four right angles, written AB ⟂ CD. In three dimensions there is a third possibility: skew lines, which neither intersect nor are parallel because they do not lie in a common plane. Skew lines appear on the test in cube and prism questions, so keep the term in mind even though this chapter is mostly about the plane.

Complementary angles have measures that add to 90°. Supplementary angles have measures that add to 180°. Neither definition requires the angles to be adjacent; two separate angles of 35° and 145° are supplementary.

Adjacent angles share a vertex and a side and have no interior points in common. A linear pair is a pair of adjacent angles whose non-shared sides are opposite rays, so together they form a straight line. The Linear Pair Postulate says the angles of a linear pair are supplementary.

When two lines intersect, they form two pairs of vertical angles, the angles that are opposite each other at the vertex. Vertical angles are congruent. The proof is short and is a favorite for proof-completion items.

Proof: Lines AC and BD intersect at E, with E between A and C and between B and D. Prove that ∠AEB ≅ ∠CED.

∠AEB and ∠BEC form a linear pair (Definition of linear pair)

m∠AEB + m∠BEC = 180° (Linear Pair Postulate)

∠BEC and ∠CED form a linear pair (Definition of linear pair)

m∠BEC + m∠CED = 180° (Linear Pair Postulate)

m∠AEB + m∠BEC = m∠BEC + m∠CED (Substitution, or transitive property of equality)

m∠AEB = m∠CED (Subtraction property of equality)

∠AEB ≅ ∠CED (Definition of congruent angles)

This completes the proof.

Memory Aid: Vertical angles "look at each other" across the vertex, like the opposite tips of an X. Linear pairs "sit side by side" on a straight line.

A transversal is a line that crosses two or more lines at different points. When a transversal t crosses lines l and m, it creates eight angles: four at each intersection. The angles between l and m are interior; the angles outside them are exterior. The pairs have names based on position.

Angle Pairs Formed by a Transversal

Same-side interior angles are also called consecutive interior angles. The Corresponding Angles Postulate (if two parallel lines are cut by a transversal, corresponding angles are congruent) is usually taken as the starting assumption, and the others are proved from it together with vertical angles and linear pairs. For example, an alternate interior angle is vertical to one angle of a corresponding pair, so it is congruent to the other angle of that pair.

When parallel lines are cut by a transversal that is not perpendicular to them, every one of the eight angles is either the acute measure a or the obtuse measure 180° − a. This single fact can solve most transversal questions in seconds: identify whether the angle you need is acute or obtuse and use a or 180° − a.

Each parallel-line theorem has a converse, and the converses are how you prove lines parallel. If two lines are cut by a transversal so that corresponding angles are congruent, or alternate interior angles are congruent, or alternate exterior angles are congruent, or same-side interior angles are supplementary, then the lines are parallel. Two more useful facts: in a plane, two lines perpendicular to the same line are parallel, and if a line is perpendicular to one of two parallel lines, it is perpendicular to the other.

Exam Tip: Read carefully whether a question gives you parallel lines and asks for an angle (use the theorem) or gives you angle measures and asks whether lines are parallel (use the converse). Proof-completion items often hinge on exactly this distinction: the reason "If two parallel lines are cut by a transversal, then alternate interior angles are congruent" cannot justify a conclusion that the lines are parallel.

Common Trap: Two angles that look equal in a figure are not automatically congruent. Unless the problem states that the lines are parallel (or gives angle measures that prove it), you cannot use any of the transversal relationships. The test convention says figures are drawn to scale unless noted, but measurements must still come from given information, not from estimating the picture.

Worked Example: Lines p and q are parallel, and transversal t crosses both. Two alternate interior angles measure (3x + 20)° and (5x − 28)°. Find x and the measure of each of the eight angles.

Alternate interior angles formed by parallel lines are congruent, so 3x + 20 = 5x − 28.

Subtract 3x from both sides: 20 = 2x − 28. Add 28: 48 = 2x, so x = 24.

Substitute: 3(24) + 20 = 92, and 5(24) − 28 = 92. The two expressions agree, which checks the work.

Every angle in the figure is either 92° or 180° − 92° = 88°. Four angles measure 92° and four measure 88°.

Result: x = 24; the angles measure 92° and 88°.

Worked Example: Lines j and k are cut by a transversal. Two same-side interior angles measure (2x + 10)° and (4x − 40)°. For what value of x are j and k parallel, and what are the angle measures then?

By the converse of the Same-Side Interior Angles Theorem, the lines are parallel exactly when the same-side interior angles are supplementary.

Notice that setting the two expressions equal (2x + 10 = 4x − 40) would give x = 25 and two 60° angles that add to 120°, not 180°. That value is a typical distractor.

Many items describe angles in words: "an angle is 30° more than twice its complement," "the supplement of an angle is four times its complement." Translate carefully. If the angle is x, its complement is 90 − x and its supplement is 180 − x. Note that only angles under 90° have complements, and the supplement of an acute angle is always exactly 90° more than its complement.

Worked Example: The supplement of an angle is four times the complement of the same angle. Find the angle.

Let the angle measure x degrees. Its supplement is 180 − x and its complement is 90 − x.

Translate the sentence: 180 − x = 4(90 − x).

Distribute: 180 − x = 360 − 4x. Add 4x and subtract 180: 3x = 180, so x = 60.

Check: the supplement is 120, the complement is 30, and 120 = 4(30).

Result: The angle measures 60°.

Worked Example: Lines AB and CD intersect at point E. The vertical angles ∠AEC and ∠BED measure (2y + 15)° and (3y − 25)°. Find m∠AED.

Vertical angles are congruent: 2y + 15 = 3y − 25, so y = 40.

∠AED forms a linear pair with ∠AEC, so m∠AED = 180° − 95° = 85°.

Result: m∠AED = 85°.

A classic problem places a point between two parallel lines and connects it to each line, forming a "bent" path. Suppose lines l and m are horizontal and parallel, with l on top. Point P lies between them. Segment PA goes up and to the left to point A on l, and segment PB goes down and to the left to point B on m, so the path A to P to B bends toward the right. If the angle between line l and segment AP, measured on the side toward P (the interior side), is 38°, and the angle between line m and segment BP on the interior side is 47°, then m∠APB = 38° + 47° = 85°.

The justification: draw a line through P parallel to l (and therefore parallel to m). This line splits ∠APB into two angles. Each one is an alternate interior angle with one of the given angles, so the parts measure 38° and 47°, and their sum is 85°. This auxiliary-line technique also proves the triangle angle-sum theorem in the next chapter, and the same idea extends to paths with several bends: the angles pointing one way add up to the angles pointing the other way.

Exam Tip: When a figure has parallel lines but the angle you need is not directly on a transversal, draw an auxiliary line through the vertex parallel to the given lines, or extend a segment until it meets one of the parallel lines. Either move creates the transversal you need.

The distance from a point to a line is the length of the perpendicular segment from the point to the line; it is the shortest segment from the point to the line. The perpendicular bisector of a segment is the line perpendicular to the segment at its midpoint. The Perpendicular Bisector Theorem states that a point lies on the perpendicular bisector of segment AB if and only if it is equidistant from A and B. You will use this to locate circumcenters in Chapter 1.2 and in constructions in Section 2. In the coordinate plane, perpendicular lines have slopes that are negative reciprocals, which Section 4 develops.

Questions on lines and angles commonly take these forms: numeric entry of an angle measure after solving a linear equation; select-all items asking which angle pairs must be congruent given that two lines are parallel; text-completion proofs where you choose the reason for a step from a drop-down list; and "which additional fact would prove l ∥ m" items. For select-all items, test each option separately against the definitions; an option that is true only in the specific drawing but not forced by the givens should be left unchecked.

Linear pairs are supplementary, and vertical angles are congruent; the vertical-angle proof uses two linear pairs and subtraction.

With parallel lines cut by a transversal, corresponding, alternate interior, and alternate exterior angles are congruent, while same-side interior and same-side exterior angles are supplementary.

Every angle formed by a transversal of two parallel lines is either a or 180° − a, so classify the angle as acute or obtuse first.

For bent paths between parallel lines, draw an auxiliary parallel line through the bend; the bend angle equals the sum of the two interior angles at the lines.

Translate word problems with x, 90 − x, and 180 − x, and check that your answer satisfies the original sentence.