Mathematics Study Guide for the ACT

Geometry: Part 2

Triangles

A triangle is a three-sided polygon with three angles. There are six types of triangles based on side length and angle measures:

• equilateral triangle—all three sides have equal length

• isosceles triangle—two sides have the same length

• scalene triangle—all three sides have different lengths

• acute triangle—all three angles measure less than \(90^{\circ}\)

• obtuse triangle—one angle measures greater than \(90^{\circ}\)

• right triangle—exactly one angle measures \(90^{\circ}\)

Right Triangles

In a right triangle, one angle measures \(90^{\circ}\). The side opposite to the right angle is called the hypotenuse and the other two sides are the legs of the triangle. We use the Pythagorean theorem to solve for unknown sides in a right triangle.

Pythagorean Theorem

The Pythagorean theorem is a useful tool that lets us solve problems involving right triangles. It helps us find the length of a missing side of a right triangle given that we know the lengths of the other two sides. It also helps us determine if a given triangle is a right triangle or not. The formula is:

\[a^2 + b^2 = c^2\]

where \(c\) is the length of the hypotenuse and \(a\) and \(b\) are the lengths of the other two sides (legs) of the triangle.

Special Triangles

There are two special right triangles with specific angle measures and side length ratios:

• \(30-60-90\) triangle—In this triangle, the three angles measure \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\). The side lengths are related by the ratio \(1:\sqrt{3}:2\).

• \(45-45-90\) triangle—In this triangle, the two acute angles both measure \(45^{\circ}\) and the remaining angle is the \(90^{\circ}\) angle. The side lengths are in the ratio \(1:1:\sqrt{2}\).

Similar and Congruent Triangles

Similar triangles are triangles that have the same shape but are not necessarily the same size. The ratios of the corresponding sides are equal, and the corresponding angles are congruent.

Congruent triangles are triangles that have the same size and shape. Corresponding angles are equal. Moreover, corresponding sides are also congruent.

Circles

A circle is a perfectly symmetrical shape with a constant radius that extends from the center to form a curved boundary. The radius of a circle is the distance from the center of the circle to any point that is on the boundary of the circle. The diameter of a circle has twice the length of the radius.

Here are some other terms you need to know when working with circles:

central angle—This is an angle whose vertex is at the center of a circle. The sides intersect the circle. The measure of a central angle is equal to the measure of the arc it intercepts.

chord—This is the line segment that connects any two points on the circumference of a circle. The diameter is the longest chord of a circle.

arc—This is part of the circumference of a circle. Arcs can be measured in degrees or lengths. They are used to describe the portion of the circle that is enclosed by two radii (plural of radius) or a chord.

Measuring Circles

Circumference

The circumference of a circle is the distance around its boundary. It is the perimeter of the circle. The formula to find the circumference is:

\[C = 2\pi r\]

where \(r\) is the radius of the circle.

Note: Remember, \(\pi\) is the universal constant approximately equal to \(3.14\).

Area

The area of a circle is the amount of space enclosed by it. The formula to find the area is:

\[A = \pi r^{2}\]

where \(r\) is the radius of the circle.

Comparing Geometric Figures

Comparing geometric figures means analyzing and comparing the properties, shapes, and sizes of them. This includes comparing the measures of angles, side lengths, areas, and volumes of geometric shapes. You can use mathematical reasoning and logic to compare geometric figures. This can involve using properties and relationships of geometric shapes, such as congruence, similarity, or proportionality, to compare their sizes, shapes, or other attributes. Another approach is to use measurement techniques, such as finding the lengths of sides or measures of angles, to compare different figures.

Angles

An angle is formed by two rays that share a common endpoint, known as the vertex of the angle. The rays are the sides of the angle. Angles are measured in degrees and describe the amount of rotation between two intersecting lines.

Angle Properties

Angle properties are characteristics and rules that describe the relationships between angles. They are used to solve problems that involve finding unknown angles. Below, we list some commonly used angle properties:

• angle sum property—The sum of the three angles in a triangle is equal to \(180^{\circ}\).

• vertical angle property—Vertical angles are pairs of opposite angles formed by two intersecting lines. This properly states that vertical angles are equal.

• corresponding angle property—Corresponding angles are formed when a pair of parallel lines are intersected by a transversal line. This property states that corresponding angles are equal in measure.

• interior angles of a polygon property—The sum of the interior angles of a polygon with \(n\) sides can be found using the formula \((n-2) \times 180\).

Special Angle Measures

Special angles are angles that have distinct properties. They are used to describe angles based on their unique features.

• right angle—This is an angle that measures \(90^{\circ}\).

• acute angle—This is an angle that measures less than \(90^{\circ}\).

• obtuse angle—This is an angle that measures greater than \(90^{\circ}\).

• straight angle—This angle measures exactly \(180^{\circ}\). It is the measure of a straight line.

• complementary angles—Two angles that add up to \(90^{\circ}\) are known as complementary angles. In the figure below, \(\angle AOC\) and \(\angle COB\) are complementary.

• supplementary angles—Two angles that add up to \(180^{\circ}\) are known as supplementary angles. In the figure below, \(\angle PRS\) and \(\angle QRS\) are supplementary.