SAT Math 13: Lines, Angles, and Triangles

Triangle

A polygon formed by three line segments connecting three noncollinear vertices. It has three sides, three vertices, and three interior angles.

Base and apex of a triangle

A chosen side is called the base, and the vertex opposite that side is the apex. The height is the shortest perpendicular distance from the apex to the base or its extension.

Isosceles, equilateral, and scalene triangles

An isosceles triangle has at least two equal sides, an equilateral triangle has three equal sides, and a scalene triangle has three sides of different lengths.

How are triangles classified by their angle measures?

A right triangle has one $90^\circ$ angle, an acute triangle has three angles less than $90^\circ$, and an obtuse triangle has one angle greater than $90^\circ$.

What is the sum of the interior angles of a Euclidean triangle?

The interior angles always sum to $180^\circ$ or $\pi$ radians.

What can be concluded if two angles of a triangle are known?

The third angle is determined by subtracting the known angles from $180^\circ$: $\gamma=180^\circ-\alpha-\beta$.

What condition on three angles is necessary for them to form a nondegenerate triangle?

Each angle must be greater than $0^\circ$ and less than $180^\circ$, and the three angles must sum to $180^\circ$.

Exterior angle theorem for a triangle

An exterior angle equals the sum of the two nonadjacent interior angles. It is supplementary to the adjacent interior angle.

How do side lengths determine the relative sizes of opposite angles in a triangle?

The longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. Equal sides are opposite equal angles.

Isosceles triangle angle theorem

The angles opposite the equal sides of an isosceles triangle are equal. Conversely, equal angles are opposite equal sides.

What are the angle measures of an equilateral triangle?

All three angles are equal, and because their sum is $180^\circ$, each angle measures $60^\circ$.

Triangle inequality

For three positive lengths to form a triangle, the sum of any two side lengths must be greater than or equal to the third. Equality produces only a degenerate, collinear triangle; a nondegenerate triangle requires strict inequality.

Can three sides of lengths $3$, $4$, and $8$ form a triangle?

No. The two shorter sides sum to $7$, which is less than $8$, violating the triangle inequality.

What is a degenerate triangle?

It is the limiting case in which all three vertices are collinear. Its side lengths satisfy equality in the triangle inequality, and its area is zero.

Vertical angles and linear pairs

Vertical angles are opposite angles formed by two intersecting lines and are congruent. A linear pair consists of adjacent angles whose measures sum to $180^\circ$.

What angle relationships are created when parallel lines are cut by a transversal?

Corresponding angles are congruent, alternate interior angles are congruent, and same-side interior angles are supplementary.

How can parallel lines be identified using a transversal?

If corresponding angles or alternate interior angles are congruent, or if same-side interior angles are supplementary, the two lines are parallel.

Perpendicular lines

Two lines are perpendicular if they intersect to form four right angles, each measuring $90^\circ$.

Perpendicular bisector and circumcenter

A perpendicular bisector passes through the midpoint of a side at a right angle. The three perpendicular bisectors meet at the circumcenter, which is equidistant from all three vertices and is the center of the circumcircle.

How does the location of a triangle's circumcenter identify the triangle's angle type?

The circumcenter is inside an acute triangle, on the midpoint of the hypotenuse of a right triangle, and outside an obtuse triangle.

Altitude and orthocenter

An altitude is a line through a vertex perpendicular to the opposite side or its extension. The three altitudes intersect at the orthocenter, which lies inside the triangle exactly when the triangle is acute.

Median and centroid

A median connects a vertex to the midpoint of the opposite side. The three medians meet at the centroid, which divides every median in a $2:1$ ratio measured from the vertex.

Angle bisector and incenter

An angle bisector divides a vertex angle into two equal angles. The three angle bisectors meet at the incenter, which is equidistant from all three sides and is the center of the incircle.

How is the area of a triangle calculated from a base and its corresponding height?

The area is $T=\frac{1}{2}bh$, where $b$ is the base length and $h$ is the perpendicular height to that base.

How can the area be found when two sides and their included angle are known?

Use $T=\frac{1}{2}ab\sin\gamma$, where $a$ and $b$ are the known sides and $\gamma$ is the angle between them.

Similarity of triangles

Similar triangles have equal corresponding angles and proportional corresponding side lengths. They have the same shape but may differ in size.

Which criteria establish triangle similarity?

AA establishes similarity from two equal angle pairs; SAS establishes it from two proportional sides and an equal included angle; SSS establishes it when all three pairs of corresponding sides are proportional.

How do corresponding lengths and areas scale in similar triangles?

If the scale factor between corresponding side lengths is $k$, then corresponding lengths such as heights and medians also scale by $k$, while areas scale by $k^2$.

Congruence of triangles

Congruent triangles have exactly the same size and shape. Corresponding sides and angles are equal, and congruent triangles are necessarily similar.

Which criteria establish triangle congruence?

The standard criteria are SSS, SAS, ASA, and AAS. Each provides enough corresponding side and angle information to determine a unique triangle.

Why does AAA establish similarity but not congruence?

Three equal angles determine shape but not scale, so triangles with the same angles can have different side lengths. Therefore AAA proves similarity, not necessarily congruence.

Why is SSA generally insufficient to prove triangle congruence?

Two sides and a non-included angle can produce two different triangles in the ambiguous case. Thus SSA is not a general congruence criterion.

Pythagorean theorem

In a right triangle with legs $a$ and $b$ and hypotenuse $c$, $a^2+b^2=c^2$. The hypotenuse is the side opposite the right angle.

Converse of the Pythagorean theorem

If the squares of the two shorter sides of a triangle sum to the square of the longest side, $a^2+b^2=c^2$, then the triangle is right.

Special right triangle: $45^\circ$-$45^\circ$-$90^\circ$

The two legs are congruent. If each leg has length $x$, the hypotenuse has length $x\sqrt{2}$, so the side ratio is $1:1:\sqrt{2}$.

Special right triangle: $30^\circ$-$60^\circ$-$90^\circ$

The side lengths opposite $30^\circ$, $60^\circ$, and $90^\circ$ are in the ratio $1:\sqrt{3}:2$. If the short leg is $x$, the long leg is $x\sqrt{3}$ and the hypotenuse is $2x$.

What side is longest in a right triangle?

The hypotenuse, which is opposite the $90^\circ$ angle, is always the longest side.