Section 10.1 Points, Lines, Planes, and Angles Math in Our World

Learning Objectives Write names for angles. Use complementary and supplementary angles to find angle measure. Use vertical angles to find angle measure. Find measures of angles formed by a transversal.

Points, Lines, and Planes The most basic geometric figures we will study are points, lines, and planes. A point is easiest thought of as a location, like a particular spot on this page. We represent points with dots, but in actuality a point has no length, width, or thickness. (We call it dimensionless.)

Points, Lines, and Planes A line is a set of connected points that has an infinite length, but no width. We draw representations of lines, but again, in actuality a line cannot be seen because it has no thickness. We will assume that lines are straight, meaning that they follow the shortest path between any two points on the line. This means that only two points are needed to describe an entire line.

Points, Lines, and Planes A plane is a two-dimensional flat surface that is infinite in length and width, but has no thickness. You might find it helpful to think of a plane as an infinitely thin piece of paper that extends infinitely far in each direction.

Points, Lines, and Planes Points and lines can be used to make other geometric figures. A line segment is a finite portion of a line consisting of two distinct points, called endpoints, and all of the points on a line between them.

Points, Lines, and Planes Any point on a line separates the line into two halves, which we call half lines. A half-line beginning at point A and continuing through point B means that A is the endpoint of the half-line, which is not included. When the endpoint of a half line is included, we call the resulting figure a ray.

Points, Lines, and Planes Below is a summary of the figures described on the last few slides showing how each is symbolized.

Angles An angle is a figure formed by two rays with a common endpoint. The rays are called the sides of the angle, and the endpoint is called the vertex. The symbol for angle is ∠, and there are a number of ways to name angles. The angle shown could be called ∠ABC, ∠CBA, ∠B, or ∠1.

Angles You should only use a single letter to denote an angle if there’s no question as to the angle represented. In this angle, ∠B is ambiguous, because there are three different angles with vertex at point B.

EXAMPLE 1 Naming Angles Name the angle shown here in four different ways. ∠RST, ∠TSR, ∠S, and ∠3. SOLUTION

Measuring Angles One way to measure an angle is in degrees, symbolized by °. One degree is defined to be 1/360 of a complete rotation.

The instrument that is used to measure an angle is called a protractor. Measuring Angles The center of the base of the protractor is placed at the vertex of the angle, and the bottom of the protractor is placed on one side of the angle. The angle measure is marked where the other side falls on the scale. m∠ABC = 40º

An acute angle has a measure between 0° and 90°. A right angle has a measure of 90°. Classifying Angles Angles can be classified by their measures.

An obtuse angle has a measure between 90° and 180°. A straight angle has a measure of 180°. Classifying Angles Angles can be classified by their measures.

Pairs of Angles Pairs of angles have various names depending on how they are related. Two angles are called adjacent angles if they have a common vertex and a common side. Adjacent angles, ∠ABC and ∠CBD are shown. The common vertex is B and the common side is the ray BC.

Pairs of Angles Pairs of angles have various names depending on how they are related. Two angles are said to be complementary if the sum of their measures is 90°. Shown are two complementary angles. The sum of the measures of ∠GHI and ∠IHJ is 90°.

Pairs of Angles Pairs of angles have various names depending on how they are related. Two angles are said to be supplementary if the sum of their measures is equal to 180°. Shown are two supplementary angles. The sum of the measures of ∠WXY and ∠YXZ is 180°.

EXAMPLE 2 Using Complementary Angles If ∠FEG and ∠GED are complementary and m∠FEG is 28°, find m∠GED. SOLUTION Since the two angles are complementary, the sum of their measures is 90°, so m∠GED + m∠FEG = 90°, and solving we get: The complement of an angle with measure 28° has measure 62°.

EXAMPLE 3 Using Supplementary Angles If ∠RQS and ∠SQP are supplementary and m∠RQS = 135°, find m∠SQP.

EXAMPLE 3 Using Supplementary Angles SOLUTION Since the two angles are supplementary, the sum of their measures is 180°, so m∠SQP + m∠RQS = 180°, and solving we get: The supplement of an angle with measure 135° has measure 45°.

EXAMPLE 4 Using Supplementary Angles If two adjacent angles are supplementary and one angle is 3 times as large as the other, find the measure of each. SOLUTION Let x = the measure of the smaller angle. The larger is three times as big, so 3x = the measure of the larger. The angles are supplementary, so their measures add to 180°. This gives us an equation: The smaller angle has measure 45°, and the larger has measure 3 x 45°, or 135°.

Pairs of Angles Just as two intersecting streets have four corners at which you can cross, when two lines intersect, four angles are formed, as we see in the figure. Angles 1 and 3 are called vertical angles, as are angles 2 and 4. It’s easy to believe from the diagram that two vertical angles have the same measure.

EXAMPLE 5 Using Vertical Angles Find m∠2, m∠3, and m∠4 when m∠1 = 40°. SOLUTION Since ∠1 and ∠3 are vertical angles and m∠1 = 40, m∠3 = 40°. Since ∠1 and ∠2 form a straight angle (180°), m∠1 + m∠2 = 180°, and Since ∠2 and ∠4 are vertical angles, m∠4 = 140°.

Parallel Lines Two lines in the same plane are called parallel if they never intersect. You might find it helpful to think of them as lines that go in the exact same direction. If two lines l1 and l2 are parallel, we write l1 || l2. When two parallel lines are intersected by a third line, we call the third line a transversal.

Parallel Lines Shown are two parallel lines cut by a transversal. The angles between the parallel lines (angles 3–6) are called interior angles, and the ones outside the parallel lines (angles 1, 2, 7, and 8) are called exterior angles.

Parallel Lines Shown are two parallel lines cut by a transversal. Alternate interior angles are the angles formed between two parallel lines on the opposite sides of the transversal that intersects the two lines. Alternate interior angles have equal measures.

Parallel Lines Shown are two parallel lines cut by a transversal. Alternate exterior angles are the opposite exterior angles formed by the transversal that intersects two parallel lines. Alternate exterior angles have equal measures.

Parallel Lines Shown are two parallel lines cut by a transversal. Corresponding angles consist of one exterior and one interior angle with no common vertex on the same side of the transversal that intersects two parallel lines. Corresponding angles have equal measures.

EXAMPLE 6 Finding Angles Formed by a Transversal Find the measures of all the angles shown when the measure of ∠2 is 50°.

EXAMPLE 6 Finding Angles Formed by a Transversal First, let’s identify the angles that have the same measure as ∠2. They are ∠3 (vertical angles), ∠6 (corresponding angles), and ∠7 (alternate exterior angles). Mark all of these as 50° on the diagram. SOLUTION

EXAMPLE 6 Finding Angles Formed by a Transversal SOLUTION Since ∠1 & ∠2 are supplementary, m∠1 = 180° – 50° = 130°. This allows us to find the remaining angles: ∠4 is a vertical angle with ∠1, so it has measure 130° as well. Now 8 is a corresponding angle with ∠4, and ∠5 is an alternate interior angle with ∠4, which means they both have measure 130° as well. 130° 130° 130° 130°