TMAT 103

Chapter 2

Review of Geometry

TMAT 103

§2.1

Angles and Lines

§2.1 – Angles and Lines

• A right angle measures 90

§2.1 – Angles and Lines

• An acute angle measures less than 90

§2.1 – Angles and Lines

• An obtuse angle measures more than 90

§2.1 – Angles and Lines

• Two vertical angles are the opposite angles formed by two intersecting lines

• Two angles are supplementary when their sum is 180

• Two angles are complementary when their sum is 90

§2.1 – Angles and Lines

• Angles p and q are vertical, as are m and n

• Angles p and n are supplementary, as are angels m and q

§2.1 – Angles and Lines

• 2 lines are perpendicular when they form a right angle

• The shortest distance between a point and a line is the perpendicular distance between them

§2.1 – Angles and Lines

• Two lines are parallel if they lie in the same plane and never intersect

• If two parallel lines are intersected by a third line (called a transversal), then– Alternate interior angles are equal– Corresponding angles are equal– Interior angles on the same side of the

transversal are supplementary

§2.1 – Angles and Lines

a and g are equal (alternate interior)a and e are equal (corresponding)a + f = 180

TMAT 103

§2.2

Triangles

§2.2 – Triangles

• A polygon is a closed figure whose sides are all line segments

• A triangle is a polygon with 3 sides

§2.2 – Triangles

• Types of triangles– Scalene – no 2 sides are equal– Isosceles – 2 sides are equal– Equilateral – all 3 sides are equal

§2.2 – Triangles

• Types of triangles– Acute – all 3 angles are acute– Obtuse – one angle is obtuse– Right – one angle is 90

§2.2 – Triangles

• In a right triangle, the side opposite the right angle is the hypotenuse, and the other two sides are the legs

• Pythagorean Theorem: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the 2 legs

222 bac

§2.2 – Triangles

• The median of a triangle is the line segment joining any vertex to the midpoint of the opposite side

§2.2 – Triangles

• The altitude of a triangle is a perpendicular line segment from any vertex to the opposite side

§2.2 – Triangles

• An angle bisector of a triangle is a line segment that bisects any angle and intersects the opposite side

§2.2 – Triangles

• The sum of the interior angles of any triangle is 180

• In a 30 – 60 – 90 triangle– The side opposite the 30 angle equals ½ the hypotenuse

– The side opposite the 60 angle equals times the length of the hypotenuse

2

3

§2.2 – Triangles

• Perimeter and Area– Perimeter – distance around

– The area of a triangle is ½ the base times the height• A = ½ bh

– Heron’s Formula• When only the 3 sides of a triangle are known

cbas

csbsassA

21 where

))()((

§2.2 – Triangles

• Triangles are similar () if their corresponding angles are equal or if their corresponding sides are in proportion

§2.2 – Triangles

• Triangles are congruent () if their corresponding angles and sides are equal

TMAT 103

§2.3

Quadrilaterals

§2.3 – Quadrilaterals

• A quadrilateral is a polygon with 4 sides

• A parallelogram is a quadrilateral having 2 pairs of parallel sides

§2.3 – Quadrilaterals

• The area of a parallelogram is the base times the height– A = bh

• The opposite sides and opposite angles of a parallelogram are equal

§2.3 – Quadrilaterals

• The diagonal of a parallelogram divides it into 2 congruent triangles

• The diagonals of a parallelogram bisect each other

§2.3 – Quadrilaterals

• A rectangle is a parallelogram with right angles

• A square is a rectangle with equal sides

• A rhombus is a parallelogram with equal sides

§2.3 – Quadrilaterals

• A trapezoid is a quadrilateral with only one pair of parallel sides

• The area of a trapezoid is given by the formula: )(2

1 bahA

TMAT 103

§2.4

Circles

§2.4 – Circles

• A circle is the set of all points on a curve equidistant from a given point called the center

• A radius is the line segment joining the center and any point on the circle

• A diameter is the chord passing through the center• A tangent is a line intersecting a circle at only one

point• A secant is a line intersecting a circle in two points• A semicircle is half of a circle

§2.4 – Circles

• Circle terminology

§2.4 – Circles

• The area of a circle is given by:– A = r2

• r is the radius

• The circumference of a circle is given by either of the following:– C = 2r

• r is the radius

– C = d• d is the diameter

§2.4 – Circles

• Circular Arcs– A central angle is formed between 2 radii and has its

vertex at the center of the circle

– An inscribed angle has vertex on the circle and sides are chords

– An arc is the part of the circle between the 2 sides of a central or inscribed angle

– The measure of an arc is equal to• the measure of the corresponding central angle

• twice the measure of the corresponding inscribed angle

§2.4 – Circles

• Example of central and inscribed angles

§2.4 – Circles

• Measurement relationships

§2.4 – Circles

• An angle inscribed in a semicircle is a right angle

§2.4 – Circles

• Find the measure of the blue arc

§2.4 – Circles

• A line tangent to a circle is perpendicular to the radius at the point of tangency

TMAT 103

§2.5

Areas and Volumes of Solids

§2.5 – Areas and Volumes of Solids

• The lateral surface area of a solid is the sum of the areas of the sides excluding the area of the bases

• The total surface area of a solid is the sum of the lateral surface area plus the area of the bases

• The volume of a solid is the number of cubic units of measurement contained in the solid

§2.5 – Areas and Volumes of Solids

• In the following figures, B = area of base, r = length of radius, and h = height