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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
c2  a2  b2
§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 23 times the length
of the hypotenuse
§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
A  s ( s  a)( s  b)( s  c)
where s 
1
2
a  b  c 
§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: A  12 h(a  b)
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 = 2 r
• 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