Geometry Essentials
87 flashcards by MentisLibrary
Angles, triangles, circles, solids, and coordinate proofs, with the formula and the reasoning behind it on every card. Learn why each theorem is true, not just what it says.
- What are the three undefined terms of geometry?
- Point, line, and plane
- What is the difference between a segment, a ray, and a line?
- A segment has two endpoints, a ray has one endpoint and continues forever, a line continues forever in both directions
- What are acute, right, obtuse, and straight angles?
- Acute is less than $90°$, right is exactly $90°$, obtuse is between $90°$ and $180°$, straight is exactly $180°$
- What are complementary and supplementary angles?
- Complementary angles sum to $90°$, supplementary angles sum to $180°$
- What are vertical angles and what is true about them?
- The opposite pairs formed by two intersecting lines. They are always congruent
- What is a transversal?
- A line that crosses two or more other lines
- If two parallel lines are cut by a transversal, what is true of corresponding angles?
- They are congruent
- What is true of alternate interior angles on parallel lines?
- They are congruent
- What is true of same-side interior angles on parallel lines?
- They are supplementary, summing to $180°$
- What is the triangle angle sum theorem?
- The interior angles of any triangle sum to $180°$
- What is the exterior angle theorem?
- An exterior angle of a triangle equals the sum of the two remote interior angles
- How are triangles classified by sides?
- Equilateral has three equal sides, isosceles has at least two, scalene has none equal
- How are triangles classified by angles?
- Acute has all angles under $90°$, right has one $90°$ angle, obtuse has one angle over $90°$
- What is the isosceles triangle theorem?
- If two sides of a triangle are congruent, the angles opposite them are congruent
- What is the triangle inequality?
- The sum of any two sides must exceed the third side
- What are the triangle congruence criteria?
- SSS, SAS, ASA, AAS, and HL for right triangles
- What does CPCTC stand for?
- Corresponding Parts of Congruent Triangles are Congruent
- What are the triangle similarity criteria?
- AA, SSS with proportional sides, and SAS with proportional sides around an equal angle
- What is the Pythagorean theorem?
- In a right triangle, $a^2 + b^2 = c^2$ where $c$ is the hypotenuse
- What is the converse of the Pythagorean theorem?
- If $a^2 + b^2 = c^2$ then the triangle is right
- What are the side ratios in a 45-45-90 triangle?
- $1 : 1 : \sqrt{2}$, so the hypotenuse is a leg times $\sqrt{2}$
- What are the side ratios in a 30-60-90 triangle?
- $1 : \sqrt{3} : 2$, short leg opposite $30°$, long leg opposite $60°$, hypotenuse opposite $90°$
- What are the common Pythagorean triples?
- $3, 4, 5$; $5, 12, 13$; $8, 15, 17$; $7, 24, 25$
- What is a midsegment of a triangle?
- A segment joining two side midpoints. It is parallel to the third side and half its length
- What is a median of a triangle and where do the medians meet?
- A segment from a vertex to the midpoint of the opposite side. The three medians meet at the centroid
- What is an altitude and where do the altitudes meet?
- A perpendicular segment from a vertex to the opposite side line. The three meet at the orthocenter
- What is the incenter of a triangle?
- The point where the three angle bisectors meet, equidistant from all three sides
- What is the circumcenter of a triangle?
- The point where the three perpendicular bisectors meet, equidistant from all three vertices
- What is the area of a triangle?
- $A = \dfrac{1}{2}bh$
- What is Heron's formula?
- $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \dfrac{a+b+c}{2}$