Geometry Essentials: Lines, Angles & Shapes

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 = 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

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