Basic concept
What is the name of a part of a line bounded by two distinct endpoints?
Line Segment
Which instrument is used to measure the degree measure of an angle?
A protractor.
State the key property of vertical angles.
Vertical angles are equal (congruent).
What are perpendicular lines?
Two lines that intersect at a right angle (90°).
[DIAGRAM: A straight line with 4 collinear points marked in order: A, B, C, D]. How many distinct line segments can be named using these endpoints?
6 line segments: AB, AC, AD, BC, BD, CD.
How many straight lines can be drawn through any two distinct points on a plane?
Exactly one line.
What is the degree measure of a straight angle?
180 degrees (180°).
One of two adjacent angles measures 47°. What is the measure of the second adjacent angle?
133° (since 180° - 47° = 133°).
Which mathematical symbol is used to denote perpendicular lines a and b?
The perpendicular symbol (a ⊥ b).
What is the maximum number of right angles that a convex quadrilateral can have?
Exactly 4 right angles (e.g., in a rectangle or square).
What is formed when a point is placed on a line, dividing it into two parts? What is each part called?
Two rays (or opposite rays).
[DIAGRAM: Straight angle AOB divided by ray OC such that angle BOC = 75°]. Find the degree measure of angle AOC.
105° (since 180° - 75° = 105°).
[DIAGRAM: Two intersecting lines form vertical angles AOC and BOD, where angle AOC = 65°]. Find the measure of angle BOD.
Angle BOD = 65° (because vertical angles are equal).
How many perpendicular lines to a given line can be drawn through a point lying ON that line?
Exactly one perpendicular line.
Three distinct straight lines intersect at a single point O. How many pairs of vertical angles are formed at this vertex?
6 pairs of vertical angles (formed by 6 rays extending from point O).
If point C lies on segment AB, how are the lengths of segments AB, AC, and CB related?
AB = AC + CB.
Inside angle AOB (130°), ray OK is drawn such that angle AOK is 30° larger than angle KOB. Find the measures of angles AOK and KOB.
Angle KOB = 50°, angle AOK = 80°. (Calculation: x + (x + 30°) = 130° ⇒ 2x = 100° ⇒ x = 50°).
Find the degree measures of all four angles formed by the intersection of two lines if the sum of two non-adjacent angles is 110°.
Two angles are 55° each (vertical), and the other two are 125° each (adjacent: 180° - 55° = 125°).
[DIAGRAM: From point M to line AB, perpendicular MP and inclined line segment MK are drawn]. Which line segment is longer and why?
Inclined segment MK is longer because in right triangle MPK, the perpendicular is a leg and the inclined segment is the hypotenuse (and the hypotenuse is always longer than a leg).
Three straight lines are drawn on a flat plane. What is the maximum number of points of intersection they can form?
3 intersection points (when the lines intersect pairwise to form a triangle).
Two rays have a common origin point but do not lie on the same line. What geometric figure do they form, and what is the common origin called?
They form an angle, and the common point is called the vertex.
Points A, B, and C lie on the same straight line. Given AB = 8 cm and BC = 3 cm, what are the two possible lengths for segment AC?
11 cm (if B is between A and C) or 5 cm (if C is between A and B).
The difference between two adjacent angles is 40°. Find the measure of both angles.
70° and 110°. (Calculation: x + (x + 40°) = 180° ⇒ 2x = 140° ⇒ x = 70°).
Two lines are drawn through the vertex of a right angle such that each is perpendicular to one of its sides. Find the angle between these two lines.
90°
A square of side length 4 cm is cut along grid lines into identical rectangles, each having a perimeter of 10 cm. What is the minimum number of such rectangles formed?
4 rectangles. (Explanation: A rectangle with perimeter 10 cm and dimensions on a 4x4 grid must measure 1 cm x 4 cm, since 2*(1 + 4) = 10 cm. The area of the square is 16 cm² and each rectangle has an area of 4 cm², so 16 / 4 = 4 rectangles).