Math 1312 - Review for Test 2

When: Wednesday, October 20. Where: In class What is covered: 2.4, 2.5, 2.6 and Chapters 3 and 4 ( the sections that were 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. ------------------------------------------------------------------------------------------------------------

Practice problems 1. Classify the triangle

a) All sides of ABC∆ are of the same length.

b) In DEF∆ , 6=DE , 6=EF , 8=DF .

c) All angles of ABC∆ measure o60 .

d) In DEF∆ , oDm 40=∠ , oEm 50=∠ , and oFm 90=∠

2. Given: Right ABC∆ with right C∠ , 47 +=∠ xAm , 25 +=∠ xBm

Find: x

3. Given: )2(81 +=∠ xm , 353 −=∠ xm , and 2)1(55 −+=∠ xm

Find: x

4. With angle measures as indicated, determine Nm∠ and Pm∠ .

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.

B

P

N

33

x

2x

1 2 3 4

5

7. Which geometric figures have symmetry with respect to a point?

8. Which words have a vertical line of symmetry?

9. It is given that DEFABC ∆≅∆

a) If oAm 37=∠ and oEm ,68=∠ find Fm∠ .

b) If cmAB 3.7= , cmBC 7.4= , and cmAC 3.6= , 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∠ , 5=CA , 12=CB

Find: AB

DAD MOM NUN EYE

12. If BCAB ≅ and oCm 69=∠ , find Am∠ . (Use the drawing below.)

13. If BCAB ≅ and oBm 40=∠ , find Am∠ . (Use the drawing below.)

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∆ , oAm 70=∠ , oBm 45=∠ , and oCm 65=∠ .

a) Is it true that AB is the longest side of ABC∆ ?

b) Is it true that BCAB <

17. In ABC∆ with right C∠ , 3=AC , 4=BC , and 5=AB . Classify the following

statement as true or false.

BmAm ∠>∠

A

B

C

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 DCAB ||

and oBm 63=∠ and oDm 118=∠ .

21. Complete the proof of the following theorem:

“In a kite, one pair of opposite angles are congruent.”

Given: Kite ABCD, ADAB ≅ and BCDC ≅

Prove: DB ∠≅∠

Statements Reasons 1. 1.

2. Draw AC . 2. Through two points, there is exactly one line

3. 3. Identity 4. ACBACD ∆≅∆ 4. 5. 5.

A

B

C

D

22. Which postulate can be used to prove that ∆ABE ≅ ∆CBE? Name the

corresponding angles and sides.

23. Find the measure of ∠U.

Hint: Find the sum of the interior angles first!! R

S T

U

V

4y°

(12y-30)° (11y-15)°

(6y)° (6y)°

A

B

C

D

E