Easy Mathematics Step-by-Step (2012)

Chapter 15. Informal Geometry

In this chapter, you learn informal geometry concepts.

Congruence

Congruent figures have exactly the same size and same shape. They will fit exactly on top of each other.

Problem Do the two figures appear to be congruent? Yes or no?

Solution

Step 1. Check whether the two figures appear to be the same size.

The two figures appear to be the same size.

Step 2. Check whether the two figures appear to have the same shape.

The two figures appear to have the same shape.

Step 3. Answer the question.

Yes, the two figures appear to be congruent.

Step 1. Check whether the two figures appear to be the same size.
The two figures do not appear to be the same size.

Step 2. Answer the question.

No, the two figures do not appear to be congruent.

Step 1. Check whether the two figures appear to be the same size.

The two figures appear to be the same size.

Step 2. Check whether the two figures appear to have the same shape.

The two figures appear to have the same shape.

Step 3. Answer the question.

Yes, the two figures appear to be congruent.

Step 1. Check whether the two figures appear to be the same size.

The two figures appear to be the same size.

Step 2. Check whether the two figures appear to have the same shape.

The two figures do not appear to have the same shape. The figure on the right appears to have all sides the same length, but the figure on the left appears to have sides that are unequal in length.

Step 3. Answer the question.

No, the two figures do not appear to be congruent.

Step 1. Check whether the two figures appear to be the same size.

The two figures appear to be the same size.

Step 2. Check whether the two figures appear to have the same shape.

The two figures appear to have the same shape.

Step 3. Answer the question.

Yes, the two figures appear to be congruent.

Similarity

Similar geometric figures have the same shape, but not necessarily the same size.

All congruent figures are also similar figures. However, not all similar figures are congruent.

Problem Do the two figures appear to be similar? Yes or no?

Solution

Step 1. Check whether the two figures appear to have the same shape.

The two figures appear to have the same shape.

Step 2. Answer the question.

Yes, the two figures appear to be similar.

Step 1. Check whether the two figures appear to have the same shape.

The two figures appear to have the same shape.

Step 2. Answer the question.

Yes, the two figures appear to be similar.

Step 1. Check whether the two figures appear to have the same shape.

The two figures do not have the same shape.

Step 2. Answer the question.

No, the two figures are not similar.

Step 1. Check whether the two figures appear to have the same shape.

The two figures appear to have the same shape.

Step 2. Answer the question.

Yes, the two figures appear to be similar.

Symmetry

Symmetry describes a characteristic of the shape of a figure or object. A figure or object has symmetry (or is symmetric) if it can be folded exactly in half resulting in two congruent halves. The line along the fold is the line of symmetry.

Problem Does the figure appear to have symmetry? Yes or no? For symmetric figures, draw a line of symmetry.

Solution

Step 1. Check whether it appears that the figure can be folded exactly in half resulting in two congruent halves.

It appears that the figure can be folded exactly in half resulting in two congruent halves.

Step 2. Answer the question.

Yes, it appears that the figure has symmetry.

Step 3. Draw a line of symmetry for the figure.

Step 1. Check whether it appears that the figure can be folded exactly in half resulting in two congruent halves.

It appears that the figure can be folded exactly in half resulting in two congruent halves.

Step 2. Answer the question.

Yes, it appears that the figure has symmetry.

Step 3. Draw a line of symmetry for the figure.

Step 1. Check whether it appears that the figure can be folded exactly in half resulting in two congruent halves.

The figure cannot be folded exactly in half resulting in two congruent halves.

Step 2. Answer the question.

No, the figure is not symmetric.

Some shapes have more than one line of symmetry.

Problem Draw all the lines of symmetry for the figure shown.

Solution

Step 1. Draw all the lines that will divide the figure into two congruent halves.

Angles

In geometry, point, line, and plane are basic terms. You might think of a point as a location in space, a line as a set of points that goes on and on in both directions, and a plane as set of points that form a flat infinite surface.

A ray is a line that extends from a point. Two rays that meet at a common point form an angle. The vertex of the angle is the point where the two rays meet, as shown in Figure 15.1.

Figure 15.1 Vertex of angle A

You use degrees (°) to measure angles. You classify angles by the number of degrees in their measurements. An acute angle measures between 0° and 90°. A right angle measures exactly 90°. An obtuse angle measures between 90° and 180°. A straight angle measures exactly 180°.

Problem Classify the angle as an acute angle, a right angle, an obtuse angle, or a straight angle.

Solution

Step 1. Note the measure of the angle.

The measure of the angle is 40°.

Step 2. Classify the angle.

40° is between 0° and 90°, so the angle is acute.

Step 1. Note the measure of the angle.

The measure of the angle is 120°.

Step 2. Classify the angle.

120° is between 0° and 180°, so the angle is obtuse.

Step 1. Note the measure of the angle.

The measure of the angle is 180°.

Step 2. Classify the angle.

The angle is a straight angle.

Step 1. Note the measure of the angle.

The measure of the angle is 90°.

Step 2. Classify the angle.

The angle is a right angle.

A right angle has a box as the angle indicator.

Lines

Lines in a plane can be parallel or intersecting. Intersecting lines cross at a point in the plane. Parallel lines (in a plane) never intersect. The distance between them is always the same. Perpendicular lines intersect at right angles.

Problem State the most specific description of the two lines shown.

Solution

Step 1. Check whether the two lines intersect.

The two lines do not intersect.

Step 2. State the most specific description of the two lines.

The two lines are parallel.

Step 1. Check whether the two lines intersect.

The two lines do intersect.

Step 2. Check whether the two lines intersect at a right angle.

The two lines do not intersect at a right angle.

Step 3. State the most specific description of the two lines.

The two lines are intersecting lines.

Step 1. Check whether the two lines intersect.

The two lines do intersect.

Step 2. Check whether the two lines intersect at a right angle.

The two lines do intersect at a right angle.

Step 3. State the most specific description of the two lines.

The two lines are perpendicular.

Polygons

A polygon is a simple, closed figure in a plane composed of sides that are straight line segments that meet only at their end points. Polygons are named by the number of sides they have. A triangle is a three-sided polygon. A quadrilateral is a four-sided polygon. A pentagon is a five-sided polygon. A hexagon is a six-sided polygon. A heptagon is a seven-sided polygon. An octagon is an eight-sided polygon. Other polygons with additional sides have special names as well. However, eventually, at a high number of sides, you simply speak of the polygon as an n-gon. If all the sides of a polygon are congruent, then the polygon is a regular polygon.

Problem Name the polygon shown.

Solution

Step 1. Count the number of sides.

Step 2. Name the polygon.

The polygon has five sides, so it is a pentagon.

Step 1. Count the number of sides.

Step 2. Name the polygon.

The polygon has eight sides, so it is an octagon.

Step 1. Count the number of sides.

Step 2. Name the polygon.

The polygon has four sides, so it is a quadrilateral.

Step 1. Count the number of sides.

Step 2. Name the polygon.

The polygon has three sides, so it is a triangle.

Step 1. Count the number of sides.

Step 2. Name the polygon.

The polygon has six sides, so it is a hexagon.

Triangles

Triangles can be classified in two different ways. You can classify triangles according to their sides as equilateral, isosceles, or scalene. An equilateral triangle has three congruent sides. An isosceles triangle has at least two congruent sides. A scalene triangle has no congruent sides.

All equilateral triangles are isosceles triangles. However, not all isosceles triangles are equilateral triangles.

Problem State the most specific name of the triangle according to its sides. (Note: Sides labeled with the same letter are congruent.)

Solution

Step 1. Count the number of congruent sides.

Step 2. State the most specific name of the triangle.

The triangle has two congruent sides, so it is an isosceles triangle.

Step 1. Count the number of congruent sides.

Step 2. State the most specific name of the triangle.

The triangle has no congruent sides, so it is a scalene triangle.

Step 1. Count the number of congruent sides.

Step 2. State the most specific name of the triangle.

The triangle has three congruent sides, so it is an equilateral triangle.

Because the sum of the angles of a triangle is 180°, the other two angles of a right triangle are acute angles.

Another way to classify triangles is according to their interior angles. The sum of the interior angles of a triangle is 180°. An acute triangle has three acute angles. A right triangle has exactly one right angle. An obtuse trianglehas exactly one obtuse angle.

Because the sum of the angles of a triangle is 180°, the other two angles of an obtuse triangle are acute angles.

Problem Name the triangle according to its angles.

Solution

Step 1. Describe the angles of the triangle.

All three angles of the triangle are acute.

Step 2. Name the triangle according to its angles.

The triangle has three acute angles, so the triangle is an acute triangle.

Step 1. Describe the angles of the triangle.

There is one right angle and two acute angles.

Step 2. Name the triangle according to its angles.

The triangle has exactly one right angle, so the triangle is a right triangle.

Step 1. Describe the angles of the triangle.

There is one obtuse angle and two acute angles.

Step 2. Name the triangle according to its angles.

The triangle has exactly one obtuse angle, so the triangle is an obtuse triangle.

Quadrilaterals

You classify quadrilaterals as either trapezoids or parallelograms. A trapezoid has exactly one pair of parallel sides. In a parallelogram, opposite sides are parallel and congruent. Note: Some texts define a trapezoid as a quadrilateral that has at least one pair of parallel sides. This situation is one of the few times that mathematicians do not agree on the definition of a term.

Some parallelograms have special names because of their special properties. A rhombus is a parallelogram that has four congruent sides. A rectangle is a parallelogram that has four interior right angles. A square is a parallelogram that has four interior right angles and four congruent sides. To be more specific, a square is a rectangle that has four congruent sides. You also can say that a square is a rhombus that has four interior right angles.

If a parallelogram has one interior right angle, then the other three interior angles also are right angles.

Do not make the mistake of thinking that squares are not rectangles. All squares are rectangles; however, not all rectangles are squares.

Problem State the most specific name of the quadrilateral. (Note: Sides labeled the same are congruent, and sides that look parallel are parallel.)

Solution

Step 1. Determine whether the quadrilateral is a trapezoid or a parallelogram.

The quadrilateral has opposite sides parallel and congruent, so it is a parallelogram.

Step 2. State the most specific name of the parallelogram.

The parallelogram has four congruent sides, so it is a rhombus.

Step 1. Determine whether the quadrilateral is a trapezoid or a parallelogram.

The quadrilateral has opposite sides parallel and congruent, so it is a parallelogram.

Step 2. State the most specific name of the parallelogram.

The parallelogram does not have a special name; it’s simply a parallelogram.

Step 1. Determine whether the quadrilateral is a trapezoid or a parallelogram.

The quadrilateral has opposite sides parallel and congruent, so it is a parallelogram.

Step 2. State the most specific name of the parallelogram.

The parallelogram has four interior right angles and four congruent sides, so it is a square.

Step 1. Determine whether the quadrilateral is a trapezoid or a parallelogram.

The quadrilateral has exactly one pair of parallel sides, so it is a trapezoid.

Step 1. Determine whether the quadrilateral is a trapezoid or a parallelogram.

The quadrilateral has opposite sides parallel and congruent, so it is a parallelogram.

Step 2. State the most specific name of the parallelogram.

The parallelogram has four interior right angles, so it is a rectangle.

Parts of a Circle

A circle is a closed figure in a plane for which all points on the figure are the same distance from a point within, called the center. A radius of a circle is a line segment joining the center of the circle to any point on the circle. The diameter is a line segment through the center of the circle with endpoints on the circle. The diameter of a circle is twice the radius. Conversely, the radius of a circle is half the diameter. See Figure 15.2.

Figure 15.2 Circle

Problem Answer the question given.

a. What is the length of the diameter of the circle shown?

b. What is the length of the radius of the circle shown?

Solution

a. What is the length of the diameter of the circle shown?

Step 1. Determine the information given.

The radius and center of the circle are shown. The radius has a length of 3.5 ft.

Step 2. Find the diameter by multiplying the length of the radius by 2.

b. What is the length of the radius of the circle shown?

Step 1. Determine the information given.

The diameter and center of the circle are shown. The diameter has a length of 4 cm.

Step 2. Find the radius by multiplying the length of the diameter by .

Solid Figures

Solid figures are three-dimensional figures that occupy space. The solid figures you should be able to recognize are prisms, pyramids, cylinders, cones, and spheres.

A prism is a solid with two congruent and parallel bases. The sides of a prism are rectangles. The bases of a prism can have the shape of any polygon. Prisms are named according to the shape of their bases. A cube is a special rectangular prism that has six congruent faces, all of which are squares.

A pyramid is a solid with exactly one base and whose sides are triangles. The base can have the shape of any polygon. Pyramids are named according to the shape of their bases.

Problem Determine whether the solid shown is a prism or a pyramid, and then state the most specific name for the solid.

Solution

Step 1. Determine whether the solid is a prism or a pyramid.

The solid has two congruent and parallel bases, and its sides are rectangles, so the solid is a prism.

Step 2. State the most specific name for the prism.

The prism has rectangular bases, so it is a rectangular prism.

Step 1. Determine whether the solid is a prism or a pyramid.

The solid has two congruent and parallel bases, and its sides are rectangles, so the solid is a prism.

Step 2. State the most specific name for the prism.

The prism has triangular bases, so it is a triangular prism.

Step 1. Determine whether the solid is a prism or a pyramid.

The solid has exactly one base, and its sides are triangles, so the solid is a pyramid.

Step 2. State the most specific name for the prism.

The pyramid has a hexagonal base, so it is a hexagonal pyramid.

A cylinder has two parallel congruent bases, which are circles. It has one rectangular side that wraps around. A cone is a three-dimensional solid that has one circular base. It has a curved side that wraps around.

A sphere is a three-dimensional figure that is shaped like a ball. Every point on the figure is the same distance from a point within, called the center of the sphere. The radius of the sphere is a line segment from the center of the sphere to any point on the sphere. The diameter of the sphere is a line segment joining two points of the sphere and passing through its center. The radius of the sphere is half the diameter. Conversely, the diameter is twice the radius. See Figure 15.3.

Figure 15.3 Sphere

Problem Determine whether the item shown is most like a cylinder, a cone, or a sphere.

Solution

Step 1. Determine whether the can is most like a cylinder, a cone, or a sphere.

The can has two parallel bases, which are circles. It has one rectangular side that wraps around, so it is most like a cylinder.

Step 1. Determine whether the snow cone cup is most like a cylinder, a cone, or a sphere.

The snow cone cup has one circular base (at the top) and a curved side that wraps around, so it is most like a cone.

Step 1. Determine whether the solid is most like a cylinder, a cone, or a sphere.

The solid has the shape of a ball, so it is a sphere.

Exercise 15

For 1–3, do the two figures appear to be congruent? Yes or no?

For 4–6, do the two figures appear to be similar? Yes or no?

For 7 and 8, if the figure has symmetry, draw all the lines of symmetry. If the figure is not symmetric, state that it is not symmetric.

For 9–11, classify the angle as an acute angle, a right angle, an obtuse angle, or a straight angle.

For 12 and 13, state the most specific description of the two lines shown.

For 14 and 15, name the polygon shown.

For 16 and 17, state the most specific name of the triangle according to its sides.

For 18–20, name the triangle according to its angles.

For 21–23, state the most specific name of the quadrilateral. (Note: Sides labeled the same are congruent, and sides that look parallel are parallel.)

24. What is the length of the diameter of the circle shown?

For 25–30, state the most specific name for the solid.