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Math 1312 - Review for Test 2
When: Friday, October 21.
Where: In class
What is covered: 2.4, 2.5, 2.6 and Chapters 3 and 4 ( sections that we covered in class)
What to bring: Picture ID, pencil, eraser, calculator (optional).
How to study: Study the class notes, solve all the problems we solved in class. Go over
the homework problems. If you have time, I also suggest solving the exercises in the
“review” part -at the end of the chapters.
Below I provided some practice problems for you. This is not a complete list, studying
only these problems is not enough!
What is covered
2.4 The angles of a Triangle
You need to know:
Terms: Scalene Triangle, Isosceles Triangle, Equilateral Triangle, Acute Triangle,
Obtuse Triangle, Right Triangle, And Equiangular Triangle.
Facts:
In a triangle, the sum of the measures of the interior angles is 180°.
Each angle of an equiangular triangle measures 60°.
The acute angles of a right triangle are complementary.
The measure of an exterior angle of a triangle equals the sum of the measures of the two
nonadjacent interior angles.
You need to be able to: Classify triangles; Find the measure of an exterior angle giving
the measures of two nonadjacent interior angles of a triangle; Use all of the above facts.
2.5 Convex Polygons
You need to know:
Terms: Polygon, Quadrilateral, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, and
Decagon.
Facts:
The sum S of the measures of the interior angles of a polygon with n sides is given by S =
(n - 2) ∗ 180°.
The measure I of each interior angle of a regular polygon of n sides is I= (n - 2) ∗ 180°/n.
The measure E of each exterior angle of a regular polygon of n sides is E=360°/n.
I+E=180°
You need to be able to: Determine the measure of an exterior and interior angle of a
regular n-gon; Find the number of sides in a regular polygon given the measure of an
exterior/interior angles.
2.6 Symmetry and Transformation
You need to know:
Terms: Line of Symmetry, Point Symmetry, Translation, Reflection, and Rotation.
3.1 Congruent Triangles
You need to know:
Terms: Congruent Triangles, SSS, SAS, ASA, and AAS.
Facts:
If the three sides of one triangle are congruent to the three sides of a second triangle, then
the triangles are congruent (SSS).
If two sides and the included angle of one triangle are congruent to two sides and the
included angle of a second triangle, then the triangles are congruent (SAS).
If two angles and the included side of one triangle are congruent to two angles and the
included side of a second triangle, then the triangles are congruent (ASA).
If two angles and a nonincluded side of one triangle are congruent to two angles and a
nonincluded side of a second triangle, then the triangles are congruent (AAS).
You need to be able to: Use SSS, SAS, ASA, AAS; Write a congruence statement.
3.2 Corresponding Parts of Congruent Triangles
You need to know:
Terms: CPCTC, Hypotenuse and Legs of a Right Triangle, HL, Pythagorean theorem.
Facts: Corresponding parts of congruent triangles are congruent (CPCTC).
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg
of a second right triangle, then the triangle are congruent (HL).
The square root of the length of the hypotenuse of a right triangle equals the sum of
squares of the lengths of the legs of the right triangle.
You need to be able to: Use all of the above facts; determine if given measures can be
sides to a right triangle.
3.3 Isosceles Triangles
You need to know:
Terms: Isosceles Triangle, Vertex, Legs, Base, Base Angles, Vertex Angle, Angle
Bisector, Median, Altitude, Perpendicular Bisector, Equilateral and Equiangular Triangle,
Perimeter.
Facts: If two sides of a triangle are congruent, then the angles opposite these sides are
also congruent.
If two angles of a triangle are congruent, then the sides opposite these angles are also
congruent.
You need to be able to: Find the measure of the vertex angle given the measure of the
base angle and vice versa.
3.5 Inequalities in a Triangle
You need to know:
Terms: SSS Inequality, SAS Inequality, and Triangle Inequality
Facts: If one side of a triangle is longer than a second side, then the measure of the angle
opposite the longer side is greater than the measure of the angle opposite the shorter side.
If the measure of one angle of a triangle is greater than the measure of a second angle,
then the side opposite the larger angle is longer than the side opposite the smaller angle.
The sum of lengths of any two sides of a triangle is greater than the length of the third
side.
The length of any side of a triangle must lie between the sum and the difference of the
lengths of the other two sides.
You need to be able to: Given measures of three angles determine the shortest, middle
and the longest sides; given measures of three sides determine the largest, middle and the
smallest angles; given measures of two sides of a triangle determine between what two
numbers must the measure of the third side lie; check if three given sides can be the sides
of a triangle.
4.1 Properties of a Parallelogram
You need to know:
Terms: Parallelogram, Diagonals of Parallelograms, and Altitudes of Parallelograms.
Facts: The opposite angles (sides) of a parallelogram are congruent.
The diagonals of a parallelogram bisect each other.
Two consecutive angles of a parallelogram are supplementary.
You need to be able to: Use all of the above facts.
4.2 The Parallelogram and Kite
You need to know:
Terms: Kite (NOT parallelogram!).
Facts:
The quadrilateral is a parallelogram if two sides are both congruent and parallel OR both
pairs of opposite sides are congruent OR diagonals bisect each other.
In a kite, ONE pair of opposite angles is congruent AND ONE diagonal is the
perpendicular bisector of the other diagonal (diagonals are always perpendicular but only
ONE diagonal is bisected by the other).
You need to be able to: Use all of the above facts.
4.3 The Rectangle, Square, and Rhombus
You need to know:
Terms: Rectangle, Square, and Rhombus.
Facts: All angles of a rectangle are right angles.
The diagonal of a rectangle are congruent.
All sides of a square and rhombus are congruent.
The diagonals of a rhombus are perpendicular.
Each diagonal of rhombus bisects a pair of opposite angles.
You need to be able to: Use all of the above facts; Answer questions like: “Is rhombus a
square and vice versa? (always, sometimes, never)”; Same question about rectangle and
square, parallelogram and square, parallelogram and rectangle, parallelogram and kite,
“In what quadrilaterals diagonals are perpendicular (bisect each other, one diagonal
bisects the other diagonal, but not both)?”;
4.4 The Trapezoid
You need to know:
Terms: Trapezoid, Bases, Legs, Base Angles (there are 2pairs!), Median, Isosceles
Trapezoid
Facts: The base angles of an isosceles trapezoid.
The diagonals of an isosceles trapezoid are congruent.
If diagonals of a trapezoid are congruent OR two base angles are congruent, the trapezoid
is an isosceles trapezoid.
The median of a trapezoid is parallel to each base and it’s length equals one-half the sum
of the lengths of the two bases.
The measures of the lower angle and the upper angle add up to 180°.
You need to be able to: Use all of the above properties; Find the length of median given
the lengths of bases of a trapezoid.
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Practice problems
1. Classify the triangle
a) All sides of ∆ABC are of the same length.
b) In ∆DEF , DE = 6 , EF = 6 , DF = 8 .
c) All angles of ∆ABC measure 60o .
d) In ∆DEF , m∠D = 40o , m∠E = 50o , and m∠F = 90o
2. Given: Right ∆ABC with right ∠C , m∠A = 7 x + 4 , m∠B = 5 x + 2
Find: x
3. Given: m∠1 = 8( x + 2) , m∠3 = 5 x − 3 , and m∠5 = 5( x + 1) − 2
Find: x
5
1 2
3
4
4. With angle measures as indicated, determine m∠N and m∠P .
P
2x
x
N
33
B
5. Find the measure of each interior angle of a regular polygon of
a) 4 sides
b) 12 sides
Name each polygon.
6. Find the measure of each exterior angle of a regular polygon of
a) 6 sides
b) 10 sides
Name each polygon.
7. Which geometric figures have symmetry with respect to a point?
8. Which words have a vertical line of symmetry?
DAD MOM NUN EYE
9. It is given that ∆ABC ≅ ∆DEF
a) If m∠A = 37 o and m∠E = 68,o find m∠F .
b) If AB = 7.3cm , BC = 4.7cm , and AC = 6.3cm , find EF .
10. Determine if the given measures can be sides to a right triangle 6, 8, 10.
11. Given: Right ∆ABC with right ∠C , CA = 5 , CB = 12
Find: AB
12. If AB ≅ BC and m∠C = 69o , find m∠A . (Use the drawing below.)
13. If AB ≅ BC and m∠B = 40o , find m∠A . (Use the drawing below.)
B
C
A
14. The measures of two sides of a triangle are 3 and 4. Between what two
numbers must the measure of the third side fall?
15. Can a triangle have sides of the following lengths: 3, 4, 7?
16. In ∆ABC , m∠A = 70o , m∠B = 45o , and m∠C = 65o .
a) Is it true that AB is the longest side of ∆ABC ?
b) Is it true that AB < BC
17. In ∆ABC with right ∠C , AC = 3 , BC = 4 , and AB = 5 . Classify the following
statement as true or false.
m∠A > m∠B
18. State whether each statement is always true, sometimes true or never true.
a) A square is a rectangle.
b) If two of the angles of a trapezoid are congruent, then the trapezoid is
isosceles.
c) The diagonals of a trapezoid bisect each other.
d) The diagonals of a parallelogram are perpendicular.
e) A rectangle is a square.
f) The diagonals of a square are perpendicular.
g) Two consecutive angles of a parallelogram are supplementary.
h) Opposite angles of a rhombus are congruent.
i) The diagonals of a rectangle are congruent.
j) The four sides of a kite are congruent.
k) The diagonals of a parallelogram are congruent.
l) The diagonals of a kite are perpendicular and bisect each other.
19. Given an isosceles trapezoid ABCD with AB parallel to DC find the length of
each diagonal if it is known that AC = 2y + 1 and BD = 19 – y.
20. Find the measures of the remaining angles of trapezoid ABCD if AB || DC
and m∠B = 63o and m∠D = 118o .
21. Complete the proof of the following theorem:
“In a kite, one pair of opposite angles are congruent.”
Given: Kite ABCD, AB ≅ AD and DC ≅ BC
Prove: ∠B ≅ ∠D
B
A
C
D
Statements
Reasons
1.
2. Draw AC .
1.
2. Through two points, there is exactly one
line
3. Identity
4.
5.
3.
4. ∆ACD ≅ ∆ACB
5.
22. Find the measure of ∠U.
V
4y°
R
(12y-30)°
(6y)°
S
(11y-15)°
U
Hint: Find the sum
of the interior
angles first!!
(6y)°
T
22. Which postulate can be used to prove that ∆ABE ≅ ∆CBE? Name the
corresponding angles and sides.
D
A
C
B
E