Find the unknown angle in triangles and quadrilaterals

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...

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...

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|...