Geometry - Triangle Types and Angles

Questions and Answers

What are the side lengths of a 30-60-90 triangle?

x-x(√3)-2x (correct)

x-√3-2x

x√3-2x

x-x

What are the side lengths of a 45-45-90 triangle?

1-√2-1

x-x-√2

√2-√2-1

1-1-√2 (correct)

The AA Similarity Postulate states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

True

The AAS Theorem states that if two angles and any side of one triangle are congruent to two angles and any side of another triangle, then the triangles are congruent.

True Signup and view all the answers

What is an acute angle?

<blockquote> 90 degrees </blockquote> Signup and view all the answers

What is an acute triangle?

A triangle with three acute angles Signup and view all the answers

What is an altitude in a triangle?

Perpendicular segment from a vertex to the line containing the opposite side Signup and view all the answers

What is an angle bisector?

A ray that divides an angle into two congruent angles Signup and view all the answers

What does the Angle Bisector Theorem state?

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle Signup and view all the answers

What is the vertex of an angle?

The common endpoint of two rays which form an angle Signup and view all the answers

What is a circumcenter?

The point of concurrency of the three perpendicular bisectors of a triangle Signup and view all the answers

What is a centroid?

The point of concurrency of the three medians of a triangle Signup and view all the answers

Define collinear points.

Points that lie on the same line Signup and view all the answers

What is the Pythagorean theorem?

a² + b² = c² Signup and view all the answers

What is a right triangle?

A triangle with one right angle Signup and view all the answers

The hypotenuse is the side opposite the right angle in a right triangle.

True Signup and view all the answers

What is an obtuse angle?

An angle between 90 and 180 degrees Signup and view all the answers

Define a median in geometry.

Bisects the opposite side into 2 congruent line segments Signup and view all the answers

What is an equilateral triangle?

All angles are congruent and all sides are the same measure Signup and view all the answers

What is the definition of a line?

A straight path that has no thickness and extends forever Signup and view all the answers

What does the SSS Similarity Theorem state?

If the lengths of the corresponding sides of two triangles are proportional, then the triangles must be similar. Signup and view all the answers

What is a transversal?

A line that crosses two or more (usually parallel) lines Signup and view all the answers

What is a tangent in terms of triangles?

Opposite/adjacent Signup and view all the answers

Define ratios in geometry.

Set up and cross-multiply, when there is an x Signup and view all the answers

What are vertical angles?

Angles opposite each other on two intersecting lines Signup and view all the answers

Study Notes

Triangle Types

30-60-90 Triangle: Side lengths are in the ratio x : x(√3) : 2x.
45-45-90 Triangle: Side lengths are in the ratio 1 : 1 : (√2).
Acute Triangle: Contains three angles each less than 90 degrees.
Obtuse Triangle: Contains one angle greater than 90 degrees.
Equilateral Triangle: All angles are congruent (60 degrees) and all sides are equal.
Isosceles Triangle: At least two sides are of equal length, which also means at least two angles are equal.
Scalene Triangle: No sides of equal length.

Angles

Acute Angle: Measures less than 90 degrees.
Obtuse Angle: Measures greater than 90 degrees but less than 180 degrees.
Right Angle: Measures exactly 90 degrees.
Exterior Angle: Formed between a side of a geometric figure and an adjacent extended side.
Interior Angle: Located between adjacent sides of a polygon.

Angle Relations and Theorems

AA Similarity Postulate: Two triangles are similar if two angles of one are congruent to two angles of the other.
AAS Theorem: Two triangles are congruent if two angles and any side of one triangle are congruent to two angles and any side of another triangle.
ASA Postulate: Two triangles are congruent if two angles and the included side of one triangle are congruent to those of another triangle.
Angle Bisector: A ray dividing an angle into two congruent angles.
Angle Bisector Theorem: A point on the bisector of an angle is equidistant from the angle's two sides.
Vertical Angles: Angles opposite each other when two lines intersect.

Triangle Congruence and Similarity

LL Theorem: Two right triangles are congruent if their legs are congruent.
HA Theorem: Two right triangles are congruent if the hypotenuse and one acute angle of one triangle match those of another triangle.
HL Theorem: Two right triangles are congruent if the hypotenuse and one leg of one match those of another triangle.
SSS Postulate: Triangles are congruent if all three sides of one triangle are congruent to all three sides of another.
SSS Similarity Theorem: Triangles are similar if the ratios of their corresponding sides are equal.

Points, Lines, and Planes

Point: Represents a location with no size, denoted by a capital letter.
Line: A straight path that extends infinitely in both directions.
Segment: A portion of a line bounded by two endpoints.
Ray: A part of a line that has one endpoint and extends infinitely in one direction.
Plane: A flat, two-dimensional surface that extends infinitely.

Properties of Shapes

Altitude: A perpendicular segment from a vertex to the line containing the opposite side.
Median: A segment that connects a vertex to the midpoint of the opposite side, dividing it into two equal segments.
Centroid: The intersection point of the three medians of a triangle.
Circumcenter: The intersection of the perpendicular bisectors of a triangle, equidistant from all vertices.

Ratios and Proportions

Ratio: A comparison of two quantities, often expressed as a fraction.
Proportions: An equation stating that two ratios are equal.
Cross Product Property: The product of the means equals the product of the extremes in a proportion.

Theorems and Proofs

Pythagorean Theorem: a² + b² = c², applies to right triangles.
Indirect Proof: Demonstrating that an assumption leads to a contradiction.
Deduction: Involves deriving specific facts from general principles.

Miscellaneous

Sine and Cosine: Trigonometric functions defined as opposite/hypotenuse and adjacent/hypotenuse, respectively.
Transversal: A line that intersects two or more other lines.
Skew Lines: Non-coplanar lines that do not intersect and are not parallel.
Venn Diagram: A graphical representation of sets and their relationships.

Geometric Relationships

Parallel Lines: Lines in a plane that never intersect.
Perpendicular Lines: Lines that intersect to form right angles.
Base and Base Angles: The bottom side of a shape and the angles formed by the intersection of bases and legs in a trapezoid.

Special Figures

Closed Figure: A shape that can be traced back to its starting point without lifting the drawing tool. Polygons are examples of closed figures.

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Description

This quiz explores various types of triangles and angles, including special right triangles like 30-60-90 and 45-45-90. It also covers properties and classifications of acute, obtuse, and scalene triangles, along with angle relationships and theorems. Test your geometry knowledge and understanding of these fundamental concepts!