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# Geometry - Triangle Types and Angles

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### 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)

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.

<p>True</p> Signup and view all the answers

### What is an acute angle?

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

### What is an acute triangle?

<p>A triangle with three acute angles</p> Signup and view all the answers

### What is an altitude in a triangle?

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

### What is an angle bisector?

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

### What does the Angle Bisector Theorem state?

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

### What is the vertex of an angle?

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

### What is a circumcenter?

<p>The point of concurrency of the three perpendicular bisectors of a triangle</p> Signup and view all the answers

### What is a centroid?

<p>The point of concurrency of the three medians of a triangle</p> Signup and view all the answers

### Define collinear points.

<p>Points that lie on the same line</p> Signup and view all the answers

### What is the Pythagorean theorem?

<p>a² + b² = c²</p> Signup and view all the answers

### What is a right triangle?

<p>A triangle with one right angle</p> Signup and view all the answers

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

<p>True</p> Signup and view all the answers

### What is an obtuse angle?

<p>An angle between 90 and 180 degrees</p> Signup and view all the answers

### Define a median in geometry.

<p>Bisects the opposite side into 2 congruent line segments</p> Signup and view all the answers

### What is an equilateral triangle?

<p>All angles are congruent and all sides are the same measure</p> Signup and view all the answers

### What is the definition of a line?

<p>A straight path that has no thickness and extends forever</p> Signup and view all the answers

### What does the SSS Similarity Theorem state?

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

### What is a transversal?

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

### Define ratios in geometry.

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

### What are vertical angles?

<p>Angles opposite each other on two intersecting lines</p> 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.
• 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!

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