Why is my implementation of reCaptcha not working?

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I'm using a combination of JavaScript, php and HTML to create a contact form, but my reCaptcha does not seem to do anything. I can hit submit anytime with or without the reCaptcha and my php code will send the mail either way.

This is a cutout of the relevant code:

<form onsubmit="return validateForm(this)" action="../structure/process_email.php" method="POST">






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<div>
</form>
https://www.google.com/recaptcha/api.js

My HTML here refers to my php which is sending my emails, but i cant seem to get the reCaptcha to be required before hitting submit.

<?php
$headers = "MIME-Version: 1.0\r\n";
$headers .= "From: xxx <xxx@xx.com\r\n";

if (isset($_POST['submit'])) {

$email_to = 'xxx@xx.com';
$email_subject = 'example subject';

$name = $_POST['name'];
$email = $_POST['email'];
$message = $_POST['message'];

$content = "From:"." ".$name."\n"."Email:"." ".$email."\n\n".$message;

mail($email_to, $email_subject, $content, $headers);
}
header('location:../contact_us/contact_us.html');
?>