1. TMAT 103 Chapter 2 Review of Geometry

2. TMAT 103 §2.1 Angles and Lines

3. §2.1 – Angles and Lines A right angle measures 90 

4. §2.1 – Angles and Lines An acute angle measures less than 90 

5. §2.1 – Angles and Lines An obtuse angle measures more than 90 

6. §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 

7. §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

8. §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

9. §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

10. §2.1 – Angles and Lines  a and  g are equal (alternate interior)  a and  e are equal (corresponding)  a +  f = 180 

11. TMAT 103 §2.2 Triangles

12. §2.2 – Triangles A polygon is a closed figure whose sides are all line segments A triangle is a polygon with 3 sides

13. §2.2 – Triangles Types of triangles –Scalene – no 2 sides are equal –Isosceles – 2 sides are equal –Equilateral – all 3 sides are equal

14. §2.2 – Triangles Types of triangles –Acute – all 3 angles are acute –Obtuse – one angle is obtuse –Right – one angle is 90 

15. §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

16. §2.2 – Triangles The median of a triangle is the line segment joining any vertex to the midpoint of the opposite side

17. §2.2 – Triangles The altitude of a triangle is a perpendicular line segment from any vertex to the opposite side

18. §2.2 – Triangles An angle bisector of a triangle is a line segment that bisects any angle and intersects the opposite side

19. §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

20. §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

21. §2.2 – Triangles Triangles are similar (  ) if their corresponding angles are equal or if their corresponding sides are in proportion

22. §2.2 – Triangles Triangles are congruent (  ) if their corresponding angles and sides are equal

23. TMAT 103 §2.3 Quadrilaterals

24. §2.3 – Quadrilaterals A quadrilateral is a polygon with 4 sides A parallelogram is a quadrilateral having 2 pairs of parallel sides

25. §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

26. §2.3 – Quadrilaterals The diagonal of a parallelogram divides it into 2 congruent triangles The diagonals of a parallelogram bisect each other

27. §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

28. §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:

29. TMAT 103 §2.4 Circles

30. §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

31. §2.4 – Circles Circle terminology

32. §2.4 – Circles The area of a circle is given by: –A =  r 2 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

33. §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

34. §2.4 – Circles Example of central and inscribed angles

35. §2.4 – Circles Measurement relationships

36. §2.4 – Circles An angle inscribed in a semicircle is a right angle

37. §2.4 – Circles Find the measure of the blue arc

38. §2.4 – Circles A line tangent to a circle is perpendicular to the radius at the point of tangency

39. TMAT 103 §2.5 Areas and Volumes of Solids

40. §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

41. §2.5 – Areas and Volumes of Solids In the following figures, B = area of base, r = length of radius, and h = height