Mathematics
Grade 10
15 min
Find the unknown angle in triangles and quadrilaterals
Find the unknown angle in triangles and quadrilaterals
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1
Introduction & Learning Objectives
Learning Objectives
Calculate the slopes of line segments to determine if sides of a polygon are parallel or perpendicular.
Apply the properties of perpendicular lines to prove the existence of right angles in triangles and quadrilaterals on the coordinate plane.
Use the distance formula to classify triangles (isosceles, equilateral) and deduce relationships between their angles.
Apply the Triangle and Quadrilateral Angle-Sum Theorems to find unknown angles after identifying a polygon's properties using coordinates.
Prove that a given set of vertices forms a specific quadrilateral (e.g., parallelogram, rectangle, square) and determine its interior angles.
Use the slopes of two intersecting lines to calculate the angle formed between them.
How do architects ensure a buil...
2
Key Concepts & Vocabulary
TermDefinitionExample
SlopeA number that measures the 'steepness' or 'incline' of a line, calculated as the change in the vertical direction (rise) divided by the change in the horizontal direction (run).The slope of a line passing through points (2, 3) and (4, 7) is m = (7-3) / (4-2) = 4 / 2 = 2.
Parallel LinesLines in a plane that never intersect. On the coordinate plane, non-vertical parallel lines have the exact same slope.A line with slope m=3 is parallel to any other line with slope m=3.
Perpendicular LinesLines that intersect to form a right angle (90°). On the coordinate plane, their slopes are negative reciprocals of each other (their product is -1).A line with slope m = 2/5 is perpendicular to a line with slope m = -5/2, because (2/5) * (-5/2) = -1.
Distance...
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Core Formulas
Slope Formula
m = \frac{y_2 - y_1}{x_2 - x_1}
Use this to calculate the slope (m) of a line segment between two points (x1, y1) and (x2, y2). It is the foundational tool for checking if sides are parallel or perpendicular.
Condition for Perpendicular Lines
m_1 \cdot m_2 = -1
If the product of the slopes of two lines is -1, the lines are perpendicular and form a 90° angle. This is the primary method for proving right angles in the coordinate plane.
Distance Formula
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Use this to find the length of a triangle or quadrilateral's sides. It is essential for identifying isosceles triangles or proving a quadrilateral is a rhombus or square.
Angle Between Two Lines
\tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|...
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Challenging
A triangle has vertices A(3, 5) and B(9, 1). Point C lies on the y-axis. Find the y-coordinate of C such that ∠ABC is a right angle.
A.2
B.5
C.3
D.-1
Challenging
The vertices of a triangle are T(0, 5), R(8, 1), and I(x, 9). If the triangle has a right angle at vertex R, what is the value of x, and what are the approximate measures of the other two angles?
A.x = 9; angles are 45°, 45°
B.x = 10; angles are 30°, 60°
C.x = 10; angles are 26.6°, 63.4°
D.x = 9; angles are 33.7°, 56.3°
Challenging
A quadrilateral has vertices A(-3, 2), B(1, 5), C(4, 1), and D(0, -2). It is proven to be a square. What is the measure of the angle formed by the intersection of its diagonals, AC and BD?
A.90°
B.45°
C.60°
D.180°
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