Triangle Congruence
What is the definition of congruent triangles?
Congruent triangles are triangles that have the same size and shape, meaning their corresponding sides and angles are equal.
What is the slope-intercept form of a linear equation?
The slope-intercept form of a linear equation is expressed as y=mx+b, where m is the slope and b is the y-intercept.
What is a system of equations?
A system of equations is a set of two or more equations with the same variables that can be solved together.
What is an arithmetic sequence?
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.
Name the postulates that can be used to prove triangles are congruent.
The postulates for proving triangles congruent include SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and HL (Hypotenuse-Leg for right triangles).
Solve for x: (2x + 5 = 15).
To solve for xx in the equation 2x+5=15, first subtract 5 from both sides to get 2x=10. Then divide both sides by 2, yielding x=5.
Solve the following system: y=2x+3 and y=−x+1.
To solve the system y=2x+3 and y=−x+1, set the equations equal to each other: 2x+3=−x+1. Solving gives x=−2/3 and substituting back gives y=1/3.
Find the 5th term of the sequence: 2, 4, 6, 8, …
The 5th term of the sequence 2, 4, 6, 8, … is 10, as it increases by 2 each time.
If two angles and a side of one triangle are equal to two angles and a side of another triangle, what can be concluded?
If two angles and a side of one triangle are equal to two angles and a side of another triangle, we can conclude that the triangles are congruent by the ASA postulate.
What does the slope of a line represent in a real-world context?
The slope of a line represents the rate of change or how steep the line is; in a real-world context, it could represent speed or growth rate.
What methods can be used to solve systems of equations?
Methods to solve systems of equations include substitution, elimination, and graphing.
What is the formula for the nth term of an arithmetic sequence?
The formula for the nth term of an arithmetic sequence is given by an=a1+(n−1)d, where a1 is the first term and d is the common difference.
How would you prove that two triangles are congruent using the SSS postulate?
To prove that two triangles are congruent using the SSS postulate, we need to show that all three sides of one triangle are equal to the three sides of the other triangle.
How can you determine if two lines are parallel from their equations?
Two lines are parallel if they have the same slope and different y-intercepts. This means their equations will be in the form y=mx+b1 and y=mx+b2 where b1≠b2.
Explain the meaning of the solution to a system of equations graphically.
Graphically, the solution to a system of equations is the point where the lines intersect, which represents the values that satisfy both equations.
Explain the difference between an arithmetic sequence and a geometric sequence.
An arithmetic sequence increases by a fixed amount (the common difference), while a geometric sequence multiplies by a constant factor (the common ratio).
If triangle ABC is congruent to triangle DEF, what can be said about their corresponding sides and angles?
If triangle ABC is congruent to triangle DEF, then side AB is equal to side DE, side BC is equal to side EF, side AC is equal to side DF, and angle A is equal to angle D, angle B is equal to angle E, and angle C is equal to angle F.
Describe how to graph the equation y=3x−4.
To graph the equation y=3x−4, start at the y-intercept −4 on the y-axis, then use the slope of 3 to rise 3 units and run 1 unit to the right for additional points.
If a system has no solution, what does that imply about the lines on a graph?
If a system has no solution, it implies that the lines are parallel and will never intersect.
Given the sequence defined by an=3n+2an=3n+2, find the 10th term.
For the sequence defined by an=3n+2, the 10th term is a10=3(10)+2 = 30+2 = 32.